diff --git a/core/modules/update/src/Form/UpdateManagerUpdate.php b/core/modules/update/src/Form/UpdateManagerUpdate.php
index fb5385c0260..7b2b680ed6b 100644
--- a/core/modules/update/src/Form/UpdateManagerUpdate.php
+++ b/core/modules/update/src/Form/UpdateManagerUpdate.php
@@ -199,42 +199,49 @@ class UpdateManagerUpdate extends FormBase {
       // Drupal core needs to be upgraded manually.
       $needs_manual = $project['project_type'] == 'core';
 
+      // If the recommended release for a contributed project is not compatible
+      // with the currently installed version of core, list that project in a
+      // separate table. To determine if the release is compatible, we inspect
+      // the 'core_compatible' key from the release info array. If it's not
+      // defined, it means we can't determine compatibility requirements (or
+      // we're looking at core), so we assume it is compatible.
+      $compatible = $recommended_release['core_compatible'] ?? TRUE;
+
       if ($needs_manual) {
-        // There are no checkboxes in the 'Manual updates' table so it will be
-        // rendered by '#theme' => 'table', not '#theme' => 'tableselect'. Since
-        // the data formats are incompatible, we convert now to the format
-        // expected by '#theme' => 'table'.
-        unset($entry['#weight']);
-        $attributes = $entry['#attributes'];
-        unset($entry['#attributes']);
-        $entry = [
-          'data' => $entry,
-        ] + $attributes;
+        $this->removeCheckboxFromRow($entry);
+        $projects['manual'][$name] = $entry;
+      }
+      elseif (!$compatible) {
+        $this->removeCheckboxFromRow($entry);
+        // If the release has a core_compatibility_message, inject it.
+        if (!empty($recommended_release['core_compatibility_message'])) {
+          // @todo In https://www.drupal.org/project/drupal/issues/3121769
+          //   refactor this into something theme-friendly so we don't have a
+          //   classless <div> here.
+          $entry['data']['recommended_version']['data']['#template'] .= ' <div>{{ core_compatibility_message }}</div>';
+          $entry['data']['recommended_version']['data']['#context']['core_compatibility_message'] = $recommended_release['core_compatibility_message'];
+        }
+        $projects['not-compatible'][$name] = $entry;
       }
       else {
         $form['project_downloads'][$name] = [
           '#type' => 'value',
           '#value' => $recommended_release['download_link'],
         ];
-      }
 
-      // Based on what kind of project this is, save the entry into the
-      // appropriate subarray.
-      switch ($project['project_type']) {
-        case 'core':
-          // Core needs manual updates at this time.
-          $projects['manual'][$name] = $entry;
-          break;
+        // Based on what kind of project this is, save the entry into the
+        // appropriate subarray.
+        switch ($project['project_type']) {
+          case 'module':
+          case 'theme':
+            $projects['enabled'][$name] = $entry;
+            break;
 
-        case 'module':
-        case 'theme':
-          $projects['enabled'][$name] = $entry;
-          break;
-
-        case 'module-disabled':
-        case 'theme-disabled':
-          $projects['disabled'][$name] = $entry;
-          break;
+          case 'module-disabled':
+          case 'theme-disabled':
+            $projects['disabled'][$name] = $entry;
+            break;
+        }
       }
     }
 
@@ -297,9 +304,45 @@ class UpdateManagerUpdate extends FormBase {
       ];
     }
 
+    if (!empty($projects['not-compatible'])) {
+      $form['not_compatible'] = [
+        '#type' => 'table',
+        '#header' => $headers,
+        '#rows' => $projects['not-compatible'],
+        '#prefix' => '<h2>' . $this->t('Not compatible') . '</h2>',
+        '#weight' => 150,
+      ];
+    }
+
     return $form;
   }
 
+  /**
+   * Prepares a row entry for use in a regular table, not a 'tableselect'.
+   *
+   * There are no checkboxes in the 'Manual updates' or 'Not compatible' tables,
+   * so they will be rendered by '#theme' => 'table', not 'tableselect'. Since
+   * the data formats are incompatible, this method converts to the format
+   * expected by '#theme' => 'table'. Generally, rows end up in the main tables
+   * that have a checkbox to allow the site admin to select which missing
+   * updates to install. This method is only used for the special case tables
+   * that have no such checkbox.
+   *
+   * @todo In https://www.drupal.org/project/drupal/issues/3121775 refactor
+   *   self::buildForm() so that we don't need this method at all.
+   *
+   * @param array[] $row
+   *   The render array for a table row.
+   */
+  protected function removeCheckboxFromRow(array &$row) {
+    unset($row['#weight']);
+    $attributes = $row['#attributes'];
+    unset($row['#attributes']);
+    $row = [
+      'data' => $row,
+    ] + $attributes;
+  }
+
   /**
    * {@inheritdoc}
    */
diff --git a/core/modules/update/tests/modules/update_test/bbb_update_test.1_1.xml b/core/modules/update/tests/modules/update_test/bbb_update_test.1_1.xml
new file mode 100644
index 00000000000..9bac7ba1f61
--- /dev/null
+++ b/core/modules/update/tests/modules/update_test/bbb_update_test.1_1.xml
@@ -0,0 +1,48 @@
+<?xml version="1.0" encoding="utf-8"?>
+<!--
+This fixture is used by Drupal\Tests\update\Functional\UpdateManagerUpdateTest.
+
+It contains 2 releases, 8.x-1.0 (which is the currently installed version) and
+8.x-1.1, which is the only available update.
+
+To ensure we've got test coverage for the case where the '<core_compatibility>'
+tag is not defined at all, this fixture does not include that value.
+-->
+<project xmlns:dc="http://purl.org/dc/elements/1.1/">
+<title>BBB Update test</title>
+<short_name>bbb_update_test</short_name>
+<dc:creator>Drupal</dc:creator>
+<supported_branches>8.x-1.</supported_branches>
+<project_status>published</project_status>
+<link>http://example.com/project/bbb_update_test</link>
+  <terms>
+   <term><name>Projects</name><value>Modules</value></term>
+  </terms>
+<releases>
+ <release>
+  <name>bbb_update_test 8.x-1.1</name>
+  <version>8.x-1.1</version>
+  <tag>8.x-1.1</tag>
+  <status>published</status>
+  <release_link>http://example.com/bbb_update_test-8-x-1-1-release</release_link>
+  <download_link>http://example.com/bbb_update_test-8.x-1.1.tar.gz</download_link>
+  <date>1250444521</date>
+  <terms>
+   <term><name>Release type</name><value>Bug fixes</value></term>
+  </terms>
+ </release>
+ <release>
+  <name>bbb_update_test 8.x-1.0</name>
+  <version>8.x-1.0</version>
+  <tag>8.x-1.0</tag>
+  <status>published</status>
+  <release_link>http://example.com/bbb_update_test-8-x-1-0-release</release_link>
+  <download_link>http://example.com/bbb_update_test-8.x-1.0.tar.gz</download_link>
+  <date>1250424521</date>
+  <terms>
+   <term><name>Release type</name><value>New features</value></term>
+   <term><name>Release type</name><value>Bug fixes</value></term>
+  </terms>
+ </release>
+</releases>
+</project>
diff --git a/core/modules/update/tests/modules/update_test/bbb_update_test.1_2.xml b/core/modules/update/tests/modules/update_test/bbb_update_test.1_2.xml
new file mode 100644
index 00000000000..f15467e689c
--- /dev/null
+++ b/core/modules/update/tests/modules/update_test/bbb_update_test.1_2.xml
@@ -0,0 +1,62 @@
+<?xml version="1.0" encoding="utf-8"?>
+<!--
+This fixture is used by Drupal\Tests\update\Functional\UpdateManagerUpdateTest.
+
+It contains 3 releases:
+- 8.x-1.0: The currently installed version in the test scenarios.
+- 8.x-1.1: An available update that does not specify '<core_compatibility>'.
+- 8.x-1.2: An available update that uses '<core_compatibility>' to require at
+    least Drupal core version 8.1.0. Since the currently installed Drupal core
+    for the tests is 8.0.0, this release will be flagged as 'Not compatible'.
+-->
+<project xmlns:dc="http://purl.org/dc/elements/1.1/">
+<title>BBB Update test</title>
+<short_name>bbb_update_test</short_name>
+<dc:creator>Drupal</dc:creator>
+<supported_branches>8.x-1.</supported_branches>
+<project_status>published</project_status>
+<link>http://example.com/project/bbb_update_test</link>
+  <terms>
+   <term><name>Projects</name><value>Modules</value></term>
+  </terms>
+<releases>
+ <release>
+  <name>bbb_update_test 8.x-1.2</name>
+  <version>8.x-1.2</version>
+  <tag>8.x-1.2</tag>
+  <core_compatibility>^8.1.0</core_compatibility>
+  <status>published</status>
+  <release_link>http://example.com/bbb_update_test-8-x-1-2-release</release_link>
+  <download_link>http://example.com/bbb_update_test-8.x-1.2.tar.gz</download_link>
+  <date>1250445521</date>
+  <terms>
+   <term><name>Release type</name><value>Bug fixes</value></term>
+  </terms>
+ </release>
+ <release>
+  <name>bbb_update_test 8.x-1.1</name>
+  <version>8.x-1.1</version>
+  <tag>8.x-1.1</tag>
+  <status>published</status>
+  <release_link>http://example.com/bbb_update_test-8-x-1-1-release</release_link>
+  <download_link>http://example.com/bbb_update_test-8.x-1.1.tar.gz</download_link>
+  <date>1250444521</date>
+  <terms>
+   <term><name>Release type</name><value>Bug fixes</value></term>
+  </terms>
+ </release>
+ <release>
+  <name>bbb_update_test 8.x-1.0</name>
+  <version>8.x-1.0</version>
+  <tag>8.x-1.0</tag>
+  <status>published</status>
+  <release_link>http://example.com/bbb_update_test-8-x-1-0-release</release_link>
+  <download_link>http://example.com/bbb_update_test-8.x-1.0.tar.gz</download_link>
+  <date>1250424521</date>
+  <terms>
+   <term><name>Release type</name><value>New features</value></term>
+   <term><name>Release type</name><value>Bug fixes</value></term>
+  </terms>
+ </release>
+</releases>
+</project>
diff --git a/core/modules/update/tests/src/Functional/UpdateManagerUpdateTest.php b/core/modules/update/tests/src/Functional/UpdateManagerUpdateTest.php
new file mode 100644
index 00000000000..3e87e9a651c
--- /dev/null
+++ b/core/modules/update/tests/src/Functional/UpdateManagerUpdateTest.php
@@ -0,0 +1,236 @@
+<?php
+
+namespace Drupal\Tests\update\Functional;
+
+/**
+ * Tests the Update Manager module's 'Update' form and functionality.
+ *
+ * @todo In https://www.drupal.org/project/drupal/issues/3117229 expand this.
+ *
+ * @group update
+ */
+class UpdateManagerUpdateTest extends UpdateTestBase {
+
+  /**
+   * Modules to enable.
+   *
+   * @var array
+   */
+  protected static $modules = [
+    'update',
+    'update_test',
+    'aaa_update_test',
+    'bbb_update_test',
+  ];
+
+  /**
+   * {@inheritdoc}
+   */
+  protected $defaultTheme = 'stark';
+
+  /**
+   * {@inheritdoc}
+   */
+  protected function setUp() {
+    parent::setUp();
+    $admin_user = $this->drupalCreateUser([
+      'administer software updates',
+      'administer site configuration',
+    ]);
+    $this->drupalLogin($admin_user);
+
+    // The installed state of the system is the same for all test cases. What
+    // varies for each test scenario is which release history fixture we fetch,
+    // which in turn changes the expected state of the UpdateManagerUpdateForm.
+    $system_info = [
+      '#all' => [
+        'version' => '8.0.0',
+      ],
+      'aaa_update_test' => [
+        'project' => 'aaa_update_test',
+        'version' => '8.x-1.0',
+        'hidden' => FALSE,
+      ],
+      'bbb_update_test' => [
+        'project' => 'bbb_update_test',
+        'version' => '8.x-1.0',
+        'hidden' => FALSE,
+      ],
+    ];
+    $this->config('update_test.settings')->set('system_info', $system_info)->save();
+  }
+
+  /**
+   * Provides data for test scenarios involving incompatible updates.
+   *
+   * These test cases rely on the following fixtures containing the following
+   * releases:
+   * - aaa_update_test.8.x-1.2.xml
+   *   - 8.x-1.2 Compatible with 8.0.0 core.
+   * - aaa_update_test.core_compatibility.8.x-1.2_8.x-2.2.xml
+   *   - 8.x-1.2 Requires 8.1.0 and above.
+   * - bbb_update_test.1_0.xml
+   *   - 8.x-1.0 is the only available release.
+   * - bbb_update_test.1_1.xml
+   *   - 8.x-1.1 is available and compatible with everything (does not define
+   *     <core_compatibility> at all).
+   * - bbb_update_test.1_2.xml
+   *   - 8.x-1.1 is available and compatible with everything (does not define
+   *     <core_compatibility> at all).
+   *   - 8.x-1.2 is available and requires Drupal 8.1.0 and above.
+   *
+   * @todo In https://www.drupal.org/project/drupal/issues/3112962:
+   *   Change the 'core_fixture' values here to use:
+   *   - '1.1' instead of '1.1-core_compatibility'.
+   *   - '1.1-alpha1' instead of '1.1-alpha1-core_compatibility'.
+   *   Delete the files:
+   *   - core/modules/update/tests/modules/update_test/drupal.1.1-alpha1-core_compatibility.xml
+   *   - core/modules/update/tests/modules/update_test/drupal.1.1-core_compatibility.xml
+   *
+   * @return array[]
+   *   Test data.
+   */
+  public function incompatibleUpdatesTableProvider() {
+    return [
+      'only one compatible' => [
+        'core_fixture' => '1.1-core_compatibility',
+        // aaa_update_test.8.x-1.2.xml has core compatibility set and will test
+        // the case where $recommended_release['core_compatible'] === TRUE in
+        // \Drupal\update\Form\UpdateManagerUpdate.
+        'a_fixture' => '8.x-1.2',
+        // Use a fixture with only a 8.x-1.0 release so BBB is up to date.
+        'b_fixture' => '1_0',
+        'compatible' => [
+          'AAA' => '8.x-1.2',
+        ],
+        'incompatible' => [],
+      ],
+      'only one incompatible' => [
+        'core_fixture' => '1.1-core_compatibility',
+        'a_fixture' => 'core_compatibility.8.x-1.2_8.x-2.2',
+        // Use a fixture with only a 8.x-1.0 release so BBB is up to date.
+        'b_fixture' => '1_0',
+        'compatible' => [],
+        'incompatible' => [
+          'AAA' => [
+            'recommended' => '8.x-1.2',
+            'range' => '8.1.0 to 8.1.1',
+          ],
+        ],
+      ],
+      'two compatible, no incompatible' => [
+        'core_fixture' => '1.1-core_compatibility',
+        'a_fixture' => '8.x-1.2',
+        // bbb_update_test.1_1.xml does not have core compatibility set and will
+        // test the case where $recommended_release['core_compatible'] === NULL
+        // in \Drupal\update\Form\UpdateManagerUpdate.
+        'b_fixture' => '1_1',
+        'compatible' => [
+          'AAA' => '8.x-1.2',
+          'BBB' => '8.x-1.1',
+        ],
+        'incompatible' => [],
+      ],
+      'two incompatible, no compatible' => [
+        'core_fixture' => '1.1-core_compatibility',
+        'a_fixture' => 'core_compatibility.8.x-1.2_8.x-2.2',
+        // bbb_update_test.1_2.xml has core compatibility set and will test the
+        // case where $recommended_release['core_compatible'] === FALSE in
+        // \Drupal\update\Form\UpdateManagerUpdate.
+        'b_fixture' => '1_2',
+        'compatible' => [],
+        'incompatible' => [
+          'AAA' => [
+            'recommended' => '8.x-1.2',
+            'range' => '8.1.0 to 8.1.1',
+          ],
+          'BBB' => [
+            'recommended' => '8.x-1.2',
+            'range' => '8.1.0 to 8.1.1',
+          ],
+        ],
+      ],
+      'one compatible, one incompatible' => [
+        'core_fixture' => '1.1-core_compatibility',
+        'a_fixture' => 'core_compatibility.8.x-1.2_8.x-2.2',
+        'b_fixture' => '1_1',
+        'compatible' => [
+          'BBB' => '8.x-1.1',
+        ],
+        'incompatible' => [
+          'AAA' => [
+            'recommended' => '8.x-1.2',
+            'range' => '8.1.0 to 8.1.1',
+          ],
+        ],
+      ],
+    ];
+  }
+
+  /**
+   * Tests the Update form for a single test scenario of incompatible updates.
+   *
+   * @dataProvider incompatibleUpdatesTableProvider
+   *
+   * @param string $core_fixture
+   *   The fixture file to use for Drupal core.
+   * @param string $a_fixture
+   *   The fixture file to use for the aaa_update_test module.
+   * @param string $b_fixture
+   *   The fixture file to use for the bbb_update_test module.
+   * @param string[] $compatible
+   *   Compatible recommended updates (if any). Keys are module identifier
+   *   ('AAA' or 'BBB') and values are the expected recommended release.
+   * @param string[][] $incompatible
+   *   Incompatible recommended updates (if any). Keys are module identifier
+   *   ('AAA' or 'BBB') and values are subarrays with the following keys:
+   *   - 'recommended': The recommended version.
+   *   - 'range': The versions of Drupal core required for that version.
+   */
+  public function testIncompatibleUpdatesTable($core_fixture, $a_fixture, $b_fixture, array $compatible, array $incompatible) {
+
+    $assert_session = $this->assertSession();
+    $compatible_table_locator = '[data-drupal-selector="edit-projects"]';
+    $incompatible_table_locator = '[data-drupal-selector="edit-not-compatible"]';
+
+    $this->refreshUpdateStatus(['drupal' => $core_fixture, 'aaa_update_test' => $a_fixture, 'bbb_update_test' => $b_fixture]);
+    $this->drupalGet('admin/reports/updates/update');
+
+    if ($compatible) {
+      // Verify the number of rows in the table.
+      $assert_session->elementsCount('css', "$compatible_table_locator tbody tr", count($compatible));
+      // We never want to see a compatibility range in the compatible table.
+      $assert_session->elementTextNotContains('css', $compatible_table_locator, 'Requires Drupal core');
+      foreach ($compatible as $module => $version) {
+        $compatible_row = "$compatible_table_locator tbody tr:contains('$module Update test')";
+        // First <td> is the checkbox, so start with td #2.
+        $assert_session->elementTextContains('css', "$compatible_row td:nth-of-type(2)", "$module Update test");
+        // Both contrib modules use 8.x-1.0 as the currently installed version.
+        $assert_session->elementTextContains('css', "$compatible_row td:nth-of-type(3)", '8.x-1.0');
+        $assert_session->elementTextContains('css', "$compatible_row td:nth-of-type(4)", $version);
+      }
+    }
+    else {
+      // Verify there is no compatible updates table.
+      $assert_session->elementNotExists('css', $compatible_table_locator);
+    }
+
+    if ($incompatible) {
+      // Verify the number of rows in the table.
+      $assert_session->elementsCount('css', "$incompatible_table_locator tbody tr", count($incompatible));
+      foreach ($incompatible as $module => $data) {
+        $incompatible_row = "$incompatible_table_locator tbody tr:contains('$module Update test')";
+        $assert_session->elementTextContains('css', "$incompatible_row td:nth-of-type(1)", "$module Update test");
+        // Both contrib modules use 8.x-1.0 as the currently installed version.
+        $assert_session->elementTextContains('css', "$incompatible_row td:nth-of-type(2)", '8.x-1.0');
+        $assert_session->elementTextContains('css', "$incompatible_row td:nth-of-type(3)", $data['recommended']);
+        $assert_session->elementTextContains('css', "$incompatible_row td:nth-of-type(3)", 'Requires Drupal core: ' . $data['range']);
+      }
+    }
+    else {
+      // Verify there is no incompatible updates table.
+      $assert_session->elementNotExists('css', $incompatible_table_locator);
+    }
+  }
+
+}