diff --git a/django/contrib/admin/static/admin/js/admin/DateTimeShortcuts.js b/django/contrib/admin/static/admin/js/admin/DateTimeShortcuts.js
index 6f71b25152bc..27223e281569 100644
--- a/django/contrib/admin/static/admin/js/admin/DateTimeShortcuts.js
+++ b/django/contrib/admin/static/admin/js/admin/DateTimeShortcuts.js
@@ -82,22 +82,32 @@
                 return;
             }
 
+            const serverTimezone =
+                document.body.dataset.adminServerTimezone || gettext("server");
             let message;
             if (timezoneOffset > 0) {
                 message = ngettext(
-                    "Note: You are %s hour ahead of server time.",
-                    "Note: You are %s hours ahead of server time.",
+                    "Note: Enter times in the %(timezone)s timezone. " +
+                        "(You are %(offset)s hour ahead.)",
+                    "Note: Enter times in the %(timezone)s timezone. " +
+                        "(You are %(offset)s hours ahead.)",
                     timezoneOffset,
                 );
             } else {
                 timezoneOffset *= -1;
                 message = ngettext(
-                    "Note: You are %s hour behind server time.",
-                    "Note: You are %s hours behind server time.",
+                    "Note: Enter times in the %(timezone)s timezone. " +
+                        "(You are %(offset)s hour behind.)",
+                    "Note: Enter times in the %(timezone)s timezone. " +
+                        "(You are %(offset)s hours behind.)",
                     timezoneOffset,
                 );
             }
-            message = interpolate(message, [timezoneOffset]);
+            message = interpolate(
+                message,
+                { timezone: serverTimezone, offset: timezoneOffset },
+                true,
+            );
 
             const warning = document.createElement("div");
             const id = inp.id;
diff --git a/django/contrib/admin/templates/admin/base.html b/django/contrib/admin/templates/admin/base.html
index 066f5fa65c3f..8ef92802a98f 100644
--- a/django/contrib/admin/templates/admin/base.html
+++ b/django/contrib/admin/templates/admin/base.html
@@ -1,4 +1,4 @@
-{% load i18n static %}<!DOCTYPE html>
+{% load i18n static tz %}<!DOCTYPE html>
 {% get_current_language as LANGUAGE_CODE %}{% get_current_language_bidi as LANGUAGE_BIDI %}
 <html lang="{{ LANGUAGE_CODE|default:"en-us" }}" dir="{{ LANGUAGE_BIDI|yesno:'rtl,ltr,auto' }}">
 <head>
@@ -25,7 +25,8 @@
 </head>
 
 <body class="{% if is_popup %}popup {% endif %}{% block bodyclass %}{% endblock %}"
-  data-admin-utc-offset="{% now "Z" %}">
+  data-admin-utc-offset="{% now "Z" %}"
+  data-admin-server-timezone="{% get_current_timezone as time_zone %}{{ time_zone }}">
 <a href="#content-start" class="skip-to-content-link">{% translate 'Skip to main content' %}</a>
 <!-- Container -->
 <div id="container">
diff --git a/django/utils/deprecation.py b/django/utils/deprecation.py
index 89c9f71e5580..f85658245483 100644
--- a/django/utils/deprecation.py
+++ b/django/utils/deprecation.py
@@ -16,7 +16,7 @@ def django_file_prefixes():
     file = getattr(django, "__file__", None)
     if file is None:
         return ()
-    return (os.path.dirname(file),)
+    return (os.path.join(os.path.dirname(file), ""),)
 
 
 class RemovedInNextVersionWarning(DeprecationWarning):
diff --git a/docs/releases/6.0.5.txt b/docs/releases/6.0.5.txt
index 91443c52aebf..1e7e1b2c3b9b 100644
--- a/docs/releases/6.0.5.txt
+++ b/docs/releases/6.0.5.txt
@@ -14,3 +14,6 @@ Bugfixes
   ``django/contrib/admin/templates/admin/change_list.html`` template added in
   Django 6.0 that could be problematic when overriding the ``pagination`` block
   (:ticket:`37029`).
+
+* Fixed a bug in Django 6.0 where deprecation warnings incorrectly skipped
+  lines from third-party packages prefixed with "django" (:ticket:`37067`).
diff --git a/js_tests/admin/DateTimeShortcuts.test.js b/js_tests/admin/DateTimeShortcuts.test.js
index 5c850ce7d249..28d07fb918a3 100644
--- a/js_tests/admin/DateTimeShortcuts.test.js
+++ b/js_tests/admin/DateTimeShortcuts.test.js
@@ -1,7 +1,15 @@
 /* global QUnit, DateTimeShortcuts */
 "use strict";
 
-QUnit.module("admin.DateTimeShortcuts");
+QUnit.module("admin.DateTimeShortcuts", {
+    afterEach: function () {
+        const $ = django.jQuery;
+        $("body")
+            .removeAttr("data-admin-server-timezone")
+            .removeAttr("data-admin-utc-offset");
+        $(".timezonewarning").remove();
+    },
+});
 
 QUnit.test("init", function (assert) {
     const $ = django.jQuery;
@@ -46,11 +54,13 @@ QUnit.test("time zone offset warning - single field", function (assert) {
         "data-admin-utc-offset",
         new Date().getTimezoneOffset() * -60 + 3600,
     );
+    $("body").attr("data-admin-server-timezone", "America/Chicago");
     DateTimeShortcuts.init();
     $("body").attr("data-admin-utc-offset", savedOffset);
     assert.equal(
         $(".timezonewarning").text(),
-        "Note: You are 1 hour behind server time.",
+        "Note: Enter times in the America/Chicago timezone. " +
+            "(You are 1 hour behind.)",
     );
     assert.equal(
         $(".timezonewarning").attr("id"),
@@ -74,6 +84,10 @@ QUnit.test("time zone offset warning - date and time field", function (assert) {
     );
     DateTimeShortcuts.init();
     $("body").attr("data-admin-utc-offset", savedOffset);
+    assert.equal(
+        $(".timezonewarning").text(),
+        "Note: Enter times in the server timezone. (You are 1 hour behind.)",
+    );
     assert.equal(
         $(".timezonewarning").attr("id"),
         "id_updated_at_timezone_warning_helptext",
diff --git a/tests/deprecation/tests.py b/tests/deprecation/tests.py
index ac610b492835..0993e8a450d5 100644
--- a/tests/deprecation/tests.py
+++ b/tests/deprecation/tests.py
@@ -1,5 +1,6 @@
 import os
 import warnings
+from pathlib import Path
 
 import django
 from django.test import SimpleTestCase
@@ -31,7 +32,11 @@ def test_with_file(self):
         prefixes = django_file_prefixes()
         self.assertIsInstance(prefixes, tuple)
         self.assertEqual(len(prefixes), 1)
-        self.assertTrue(prefixes[0].endswith(f"{os.path.sep}django"))
+        self.assertTrue(prefixes[0].endswith(f"{os.path.sep}django{os.path.sep}"))
+
+    def test_does_not_match_packages_prefixed_with_django(self):
+        other_file = Path(django.__file__).parent.parent / "djangoextra" / "__init__.py"
+        self.assertFalse(str(other_file).startswith(django_file_prefixes()))
 
 
 class RenameManagerMethods(RenameMethodsBase):