diff --git a/CHANGES b/CHANGES
index 035a4f2a496..40fa83b447b 100644
--- a/CHANGES
+++ b/CHANGES
@@ -43,6 +43,9 @@ Features added
   text
 * #9176: i18n: Emit a debug message if message catalog file not found under
   :confval:`locale_dirs`
+* #9016: linkcheck: Support checking anchors on github.com
+* #9016: linkcheck: Add a new event :event:`linkcheck-process-uri` to modify
+  URIs before checking hyperlinks
 * #1874: py domain: Support union types using ``|`` in info-field-list
 * #9268: py domain: :confval:`python_use_unqualified_type_names` supports type
   field in info-field-list
diff --git a/doc/extdev/appapi.rst b/doc/extdev/appapi.rst
index 41318e9d62b..ae81fc5628f 100644
--- a/doc/extdev/appapi.rst
+++ b/doc/extdev/appapi.rst
@@ -384,6 +384,14 @@ Here is a more detailed list of these events.
    .. versionchanged:: 1.3
       The return value can now specify a template name.
 
+.. event:: linkcheck-process-uri (app, uri)
+
+   Emitted when the linkcheck builder collects hyperlinks from document.  *uri*
+   is a collected URI.  The event handlers can modify the URI by returning a
+   string.
+
+   .. versionadded:: 4.1
+
 .. event:: build-finished (app, exception)
 
    Emitted when a build has finished, before Sphinx exits, usually used for
diff --git a/sphinx/builders/linkcheck.py b/sphinx/builders/linkcheck.py
index a46b80c081b..a635e79a98e 100644
--- a/sphinx/builders/linkcheck.py
+++ b/sphinx/builders/linkcheck.py
@@ -21,7 +21,7 @@
 from threading import Thread
 from typing import (Any, Dict, Generator, List, NamedTuple, Optional, Pattern, Set, Tuple,
                     Union, cast)
-from urllib.parse import unquote, urlparse
+from urllib.parse import unquote, urlparse, urlunparse
 
 from docutils import nodes
 from docutils.nodes import Element
@@ -627,6 +627,10 @@ def run(self, **kwargs: Any) -> None:
             if 'refuri' not in refnode:
                 continue
             uri = refnode['refuri']
+            newuri = self.app.emit_firstresult('linkcheck-process-uri', uri)
+            if newuri:
+                uri = newuri
+
             lineno = get_node_line(refnode)
             uri_info = Hyperlink(uri, self.env.docname, lineno)
             if uri not in hyperlinks:
@@ -636,12 +640,31 @@ def run(self, **kwargs: Any) -> None:
         for imgnode in self.document.traverse(nodes.image):
             uri = imgnode['candidates'].get('?')
             if uri and '://' in uri:
+                newuri = self.app.emit_firstresult('linkcheck-process-uri', uri)
+                if newuri:
+                    uri = newuri
+
                 lineno = get_node_line(imgnode)
                 uri_info = Hyperlink(uri, self.env.docname, lineno)
                 if uri not in hyperlinks:
                     hyperlinks[uri] = uri_info
 
 
+def rewrite_github_anchor(app: Sphinx, uri: str) -> Optional[str]:
+    """Rewrite anchor name of the hyperlink to github.com
+
+    The hyperlink anchors in github.com are dynamically generated.  This rewrites
+    them before checking and makes them comparable.
+    """
+    parsed = urlparse(uri)
+    if parsed.hostname == "github.com" and parsed.fragment:
+        prefixed = parsed.fragment.startswith('user-content-')
+        if not prefixed:
+            fragment = f'user-content-{parsed.fragment}'
+            return urlunparse(parsed._replace(fragment=fragment))
+    return None
+
+
 def setup(app: Sphinx) -> Dict[str, Any]:
     app.add_builder(CheckExternalLinksBuilder)
     app.add_post_transform(HyperlinkCollector)
@@ -658,6 +681,9 @@ def setup(app: Sphinx) -> Dict[str, Any]:
     app.add_config_value('linkcheck_anchors_ignore', ["^!"], None)
     app.add_config_value('linkcheck_rate_limit_timeout', 300.0, None)
 
+    app.add_event('linkcheck-process-uri')
+    app.connect('linkcheck-process-uri', rewrite_github_anchor)
+
     return {
         'version': 'builtin',
         'parallel_read_safe': True,
diff --git a/tests/roots/test-linkcheck/links.txt b/tests/roots/test-linkcheck/links.txt
index b389414c90c..c21968250d7 100644
--- a/tests/roots/test-linkcheck/links.txt
+++ b/tests/roots/test-linkcheck/links.txt
@@ -13,6 +13,8 @@ Some additional anchors to exercise ignore code
 * `Complete nonsense <https://localhost:7777/doesnotexist>`_
 * `Example valid local file <conf.py>`_
 * `Example invalid local file <path/to/notfound>`_
+* https://github.com/sphinx-doc/sphinx#documentation
+* https://github.com/sphinx-doc/sphinx#user-content-testing
 
 .. image:: https://www.google.com/image.png
 .. figure:: https://www.google.com/image2.png
diff --git a/tests/test_build_linkcheck.py b/tests/test_build_linkcheck.py
index fd7a5482abd..0d24c1dde52 100644
--- a/tests/test_build_linkcheck.py
+++ b/tests/test_build_linkcheck.py
@@ -65,8 +65,8 @@ def test_defaults_json(app):
                  "info"]:
         assert attr in row
 
-    assert len(content.splitlines()) == 10
-    assert len(rows) == 10
+    assert len(content.splitlines()) == 12
+    assert len(rows) == 12
     # the output order of the rows is not stable
     # due to possible variance in network latency
     rowsby = {row["uri"]: row for row in rows}
@@ -87,7 +87,7 @@ def test_defaults_json(app):
     assert dnerow['uri'] == 'https://localhost:7777/doesnotexist'
     assert rowsby['https://www.google.com/image2.png'] == {
         'filename': 'links.txt',
-        'lineno': 18,
+        'lineno': 20,
         'status': 'broken',
         'code': 0,
         'uri': 'https://www.google.com/image2.png',
@@ -101,6 +101,10 @@ def test_defaults_json(app):
     # images should fail
     assert "Not Found for url: https://www.google.com/image.png" in \
         rowsby["https://www.google.com/image.png"]["info"]
+    # The anchor of the URI for github.com is automatically modified
+    assert 'https://github.com/sphinx-doc/sphinx#documentation' not in rowsby
+    assert 'https://github.com/sphinx-doc/sphinx#user-content-documentation' in rowsby
+    assert 'https://github.com/sphinx-doc/sphinx#user-content-testing' in rowsby
 
 
 @pytest.mark.sphinx(