diff --git a/news/12939.trivial.rst b/news/12939.trivial.rst
new file mode 100644
index 00000000000..c9acd4f8c54
--- /dev/null
+++ b/news/12939.trivial.rst
@@ -0,0 +1 @@
+Update non PEP 440 wheel filename deprecation notice.
diff --git a/src/pip/_internal/models/wheel.py b/src/pip/_internal/models/wheel.py
index 63db978f1f5..67318ae55a8 100644
--- a/src/pip/_internal/models/wheel.py
+++ b/src/pip/_internal/models/wheel.py
@@ -6,6 +6,13 @@
 from typing import Dict, Iterable, List
 
 from pip._vendor.packaging.tags import Tag
+from pip._vendor.packaging.utils import (
+    InvalidVersion,
+    parse_wheel_filename,
+)
+from pip._vendor.packaging.utils import (
+    InvalidWheelFilename as PackagingInvalidWheelName,
+)
 
 from pip._internal.exceptions import InvalidWheelFilename
 from pip._internal.utils.deprecation import deprecated
@@ -32,20 +39,40 @@ def __init__(self, filename: str) -> None:
         self.name = wheel_info.group("name").replace("_", "-")
         _version = wheel_info.group("ver")
         if "_" in _version:
-            deprecated(
-                reason=(
-                    f"Wheel filename {filename!r} uses an invalid filename format, "
-                    f"as the version part {_version!r} is not correctly normalised, "
-                    "and contains an underscore character. Future versions of pip may "
-                    "fail to recognise this wheel."
-                ),
-                replacement=(
-                    "rename the wheel to use a correctly normalised version part "
-                    "(this may require updating the version in the project metadata)"
-                ),
-                gone_in="25.1",
-                issue=12914,
-            )
+            try:
+                parse_wheel_filename(filename)
+            except InvalidVersion as e:
+                deprecated(
+                    reason=(
+                        f"Wheel filename version part {_version!r} is not correctly "
+                        "normalised, and contained an underscore character in the "
+                        "version part. Future versions of pip will fail to recognise "
+                        f"this wheel and report the error: {e.args[0]}."
+                    ),
+                    replacement=(
+                        "rename the wheel to use a correctly normalised "
+                        "version part (this may require updating the version "
+                        "in the project metadata)"
+                    ),
+                    gone_in="25.1",
+                    issue=12938,
+                )
+            except PackagingInvalidWheelName as e:
+                deprecated(
+                    reason=(
+                        f"The wheel filename {filename!r} is not correctly normalised. "
+                        "Future versions of pip will fail to recognise this wheel. "
+                        f"and report the error: {e.args[0]}."
+                    ),
+                    replacement=(
+                        "rename the wheel to use a correctly normalised "
+                        "name (this may require updating the version in "
+                        "the project metadata)"
+                    ),
+                    gone_in="25.1",
+                    issue=12938,
+                )
+
             _version = _version.replace("_", "-")
 
         self.version = _version