diff --git a/CHANGES.rst b/CHANGES.rst
index 471d99626f0..73cb7409de0 100644
--- a/CHANGES.rst
+++ b/CHANGES.rst
@@ -53,6 +53,10 @@ Bugs fixed
 * #11715: Apply ``tls_verify`` and ``tls_cacerts`` config to
   ``ImageDownloader``.
   Patch by Nick Touran.
+* #11433: Added the ``linkcheck_allow_unauthorized`` configuration option.
+  Set this option to ``False`` to report HTTP 401 (unauthorized) server
+  responses as broken.
+  Patch by James Addison.
 
 Testing
 -------
diff --git a/doc/usage/configuration.rst b/doc/usage/configuration.rst
index f2367a8e54c..5eff8eb5728 100644
--- a/doc/usage/configuration.rst
+++ b/doc/usage/configuration.rst
@@ -2915,6 +2915,18 @@ Options for the linkcheck builder
 
    .. versionadded:: 4.4
 
+.. confval:: linkcheck_allow_unauthorized
+
+   When a webserver responds with an HTTP 401 (unauthorized) response, the
+   current default behaviour of Sphinx is to treat the link as "working".  To
+   change that behaviour, set this option to ``False``.
+
+   The default value for this option will be changed in Sphinx 8.0; from that
+   version onwards, HTTP 401 responses to checked hyperlinks will be treated
+   as "broken" by default.
+
+   .. versionadded:: 7.3
+
 
 Options for the XML builder
 ---------------------------
diff --git a/sphinx/builders/linkcheck.py b/sphinx/builders/linkcheck.py
index 460598ef24e..85351b948db 100644
--- a/sphinx/builders/linkcheck.py
+++ b/sphinx/builders/linkcheck.py
@@ -7,6 +7,7 @@
 import re
 import socket
 import time
+import warnings
 from html.parser import HTMLParser
 from os import path
 from queue import PriorityQueue, Queue
@@ -18,6 +19,7 @@
 from requests.exceptions import ConnectionError, HTTPError, SSLError, TooManyRedirects
 
 from sphinx.builders.dummy import DummyBuilder
+from sphinx.deprecation import RemovedInSphinx80Warning
 from sphinx.locale import __
 from sphinx.transforms.post_transforms import SphinxPostTransform
 from sphinx.util import encode_uri, logging, requests
@@ -66,6 +68,15 @@ def init(self) -> None:
         # set a timeout for non-responding servers
         socket.setdefaulttimeout(5.0)
 
+        if not self.config.linkcheck_allow_unauthorized:
+            deprecation_msg = (
+                "The default value for 'linkcheck_allow_unauthorized' will change "
+                "from `True` in Sphinx 7.3+ to `False`, meaning that HTTP 401 "
+                "unauthorized responses will be reported as broken by default. "
+                "See https://github.com/sphinx-doc/sphinx/issues/11433 for details."
+            )
+            warnings.warn(deprecation_msg, RemovedInSphinx80Warning, stacklevel=1)
+
     def finish(self) -> None:
         checker = HyperlinkAvailabilityChecker(self.config)
         logger.info('')
@@ -283,6 +294,7 @@ def __init__(self, config: Config,
         self.allowed_redirects = config.linkcheck_allowed_redirects
         self.retries: int = config.linkcheck_retries
         self.rate_limit_timeout = config.linkcheck_rate_limit_timeout
+        self._allow_unauthorized = config.linkcheck_allow_unauthorized
 
         self.user_agent = config.user_agent
         self.tls_verify = config.tls_verify
@@ -437,9 +449,30 @@ def _check_uri(self, uri: str, hyperlink: Hyperlink) -> tuple[str, str, int]:
             except HTTPError as err:
                 error_message = str(err)
 
-                # Unauthorised: the reference probably exists
+                # Unauthorized: the client did not provide required credentials
                 if status_code == 401:
-                    return 'working', 'unauthorized', 0
+                    if self._allow_unauthorized:
+                        deprecation_msg = (
+                            "\n---\n"
+                            "The linkcheck builder encountered an HTTP 401 "
+                            "(unauthorized) response, and will report it as "
+                            "'working' in this version of Sphinx to maintain "
+                            "backwards-compatibility."
+                            "\n"
+                            "This logic will change in Sphinx 8.0 which will "
+                            "report the hyperlink as 'broken'."
+                            "\n"
+                            "To explicitly continue treating unauthorized "
+                            "hyperlink responses as 'working', set the "
+                            "'linkcheck_allow_unauthorized' config option to "
+                            "``True``."
+                            "\n"
+                            "See sphinx-doc/sphinx#11433 for details."
+                            "\n---"
+                        )
+                        warnings.warn(deprecation_msg, RemovedInSphinx80Warning, stacklevel=1)
+                    status = 'working' if self._allow_unauthorized else 'broken'
+                    return status, 'unauthorized', 0
 
                 # Rate limiting; back-off if allowed, or report failure otherwise
                 if status_code == 429:
@@ -625,6 +658,7 @@ def setup(app: Sphinx) -> dict[str, Any]:
     app.add_config_value('linkcheck_anchors_ignore', ['^!'], '')
     app.add_config_value('linkcheck_anchors_ignore_for_url', (), '', (tuple, list))
     app.add_config_value('linkcheck_rate_limit_timeout', 300.0, '')
+    app.add_config_value('linkcheck_allow_unauthorized', True, '')
 
     app.add_event('linkcheck-process-uri')
 
diff --git a/tests/test_build_linkcheck.py b/tests/test_build_linkcheck.py
index 2ba817579d4..28529d9b0cb 100644
--- a/tests/test_build_linkcheck.py
+++ b/tests/test_build_linkcheck.py
@@ -348,8 +348,12 @@ class CustomHandler(http.server.BaseHTTPRequestHandler):
 
         def authenticated(method):
             def method_if_authenticated(self):
-                if (expected_token is None
-                        or self.headers["Authorization"] == f"Basic {expected_token}"):
+                if expected_token is None:
+                    return method(self)
+                elif not self.headers["Authorization"]:
+                    self.send_response(401, "Unauthorized")
+                    self.end_headers()
+                elif self.headers["Authorization"] == f"Basic {expected_token}":
                     return method(self)
                 else:
                     self.send_response(403, "Forbidden")
@@ -392,6 +396,21 @@ def test_auth_header_uses_first_match(app):
     assert content["status"] == "working"
 
 
+@pytest.mark.filterwarnings('ignore::sphinx.deprecation.RemovedInSphinx80Warning')
+@pytest.mark.sphinx(
+    'linkcheck', testroot='linkcheck-localserver', freshenv=True,
+    confoverrides={'linkcheck_allow_unauthorized': False})
+def test_unauthorized_broken(app):
+    with http_server(custom_handler(valid_credentials=("user1", "password"))):
+        app.build()
+
+    with open(app.outdir / "output.json", encoding="utf-8") as fp:
+        content = json.load(fp)
+
+    assert content["info"] == "unauthorized"
+    assert content["status"] == "broken"
+
+
 @pytest.mark.sphinx(
     'linkcheck', testroot='linkcheck-localserver', freshenv=True,
     confoverrides={'linkcheck_auth': [(r'^$', ('user1', 'password'))]})
@@ -402,10 +421,9 @@ def test_auth_header_no_match(app):
     with open(app.outdir / "output.json", encoding="utf-8") as fp:
         content = json.load(fp)
 
-    # TODO: should this test's webserver return HTTP 401 here?
-    # https://github.com/sphinx-doc/sphinx/issues/11433
-    assert content["info"] == "403 Client Error: Forbidden for url: http://localhost:7777/"
-    assert content["status"] == "broken"
+    # This link is considered working based on the default linkcheck_allow_unauthorized=true
+    assert content["info"] == "unauthorized"
+    assert content["status"] == "working"
 
 
 @pytest.mark.sphinx(