diff --git a/news/9188.bugfix.rst b/news/9188.bugfix.rst
new file mode 100644
index 00000000000..a20a2600cc7
--- /dev/null
+++ b/news/9188.bugfix.rst
@@ -0,0 +1 @@
+Allow legacy versions when verifying wheel version from file name.
diff --git a/src/pip/_internal/resolution/resolvelib/candidates.py b/src/pip/_internal/resolution/resolvelib/candidates.py
index 035e118d022..5ae92d26910 100644
--- a/src/pip/_internal/resolution/resolvelib/candidates.py
+++ b/src/pip/_internal/resolution/resolvelib/candidates.py
@@ -5,6 +5,7 @@
 from pip._vendor.packaging.specifiers import InvalidSpecifier, SpecifierSet
 from pip._vendor.packaging.utils import canonicalize_name
 from pip._vendor.packaging.version import Version, _BaseVersion
+from pip._vendor.packaging.version import parse as parse_version
 from pip._vendor.pkg_resources import Distribution
 
 from pip._internal.exceptions import HashError, MetadataInconsistent
@@ -277,7 +278,7 @@ def __init__(
             assert name == wheel_name, f"{name!r} != {wheel_name!r} for wheel"
             # Version may not be present for PEP 508 direct URLs
             if version is not None:
-                wheel_version = Version(wheel.version)
+                wheel_version = parse_version(wheel.version)
                 assert version == wheel_version, "{!r} != {!r} for wheel {}".format(
                     version, wheel_version, name
                 )