diff --git a/yt_dlp/YoutubeDL.py b/yt_dlp/YoutubeDL.py
index 5bfab3a7d..9b391b278 100644
--- a/yt_dlp/YoutubeDL.py
+++ b/yt_dlp/YoutubeDL.py
@@ -705,9 +705,8 @@ def process_color_policy(stream):
         if system_deprecation:
             self.deprecated_feature(system_deprecation.replace('\n', '\n                    '))
 
-        if self.params.get('update_self') is None:
-            version_age_warning = _get_version_age_warning()
-            if version_age_warning:
+        if self.params.get('warn_when_outdated'):
+            if version_age_warning := _get_version_age_warning():
                 self.report_warning(version_age_warning)
 
         if self.params.get('allow_unplayable_formats'):
diff --git a/yt_dlp/__init__.py b/yt_dlp/__init__.py
index 4b63ed63a..8b8e22b4b 100644
--- a/yt_dlp/__init__.py
+++ b/yt_dlp/__init__.py
@@ -974,7 +974,7 @@ def parse_options(argv=None):
         '_warnings': warnings,
         '_deprecation_warnings': deprecation_warnings,
         'compat_opts': opts.compat_opts,
-        'update_self': opts.update_self,
+        'warn_when_outdated': opts.update_self is None,
     })