--- a/src/urllib3/contrib/pyopenssl.py
+++ b/src/urllib3/contrib/pyopenssl.py
@@ -69,9 +69,11 @@
 _openssl_versions: dict[int, int] = {
     util.ssl_.PROTOCOL_TLS: OpenSSL.SSL.SSLv23_METHOD,  # type: ignore[attr-defined]
     util.ssl_.PROTOCOL_TLS_CLIENT: OpenSSL.SSL.SSLv23_METHOD,  # type: ignore[attr-defined]
-    ssl.PROTOCOL_TLSv1: OpenSSL.SSL.TLSv1_METHOD,
 }
 
+if hasattr(ssl, "PROTOCOL_TLSv1") and hasattr(OpenSSL.SSL, "TLSv1_METHOD"):
+    _openssl_versions[ssl.PROTOCOL_TLSv1] = OpenSSL.SSL.TLSv1_METHOD
+
 if hasattr(ssl, "PROTOCOL_TLSv1_1") and hasattr(OpenSSL.SSL, "TLSv1_1_METHOD"):
     _openssl_versions[ssl.PROTOCOL_TLSv1_1] = OpenSSL.SSL.TLSv1_1_METHOD
 
--- a/test/test_ssl.py
+++ b/test/test_ssl.py
@@ -173,15 +173,15 @@
         "kwargs",
         [
             {
-                "ssl_version": ssl.PROTOCOL_TLSv1,
+                "ssl_version": getattr(ssl, "PROTOCOL_TLSv1", getattr(ssl, "PROTOCOL_TLSv1_2", 1)),
                 "ssl_minimum_version": ssl.TLSVersion.MINIMUM_SUPPORTED,
             },
             {
-                "ssl_version": ssl.PROTOCOL_TLSv1,
+                "ssl_version": getattr(ssl, "PROTOCOL_TLSv1", getattr(ssl, "PROTOCOL_TLSv1_2", 1)),
                 "ssl_maximum_version": ssl.TLSVersion.TLSv1,
             },
             {
-                "ssl_version": ssl.PROTOCOL_TLSv1,
+                "ssl_version": getattr(ssl, "PROTOCOL_TLSv1", getattr(ssl, "PROTOCOL_TLSv1_2", 1)),
                 "ssl_minimum_version": ssl.TLSVersion.MINIMUM_SUPPORTED,
                 "ssl_maximum_version": ssl.TLSVersion.MAXIMUM_SUPPORTED,
             },
@@ -222,10 +222,10 @@
     @pytest.mark.parametrize(
         "kwargs",
         [
-            {"ssl_version": ssl.PROTOCOL_TLSv1, "ssl_minimum_version": None},
-            {"ssl_version": ssl.PROTOCOL_TLSv1, "ssl_maximum_version": None},
+            {"ssl_version": getattr(ssl, "PROTOCOL_TLSv1", getattr(ssl, "PROTOCOL_TLSv1_2", 1)), "ssl_minimum_version": None},
+            {"ssl_version": getattr(ssl, "PROTOCOL_TLSv1", getattr(ssl, "PROTOCOL_TLSv1_2", 1)), "ssl_maximum_version": None},
             {
-                "ssl_version": ssl.PROTOCOL_TLSv1,
+                "ssl_version": getattr(ssl, "PROTOCOL_TLSv1", getattr(ssl, "PROTOCOL_TLSv1_2", 1)),
                 "ssl_minimum_version": None,
                 "ssl_maximum_version": None,
             },
--- a/test/test_util.py
+++ b/test/test_util.py
@@ -982,9 +982,11 @@
     @pytest.mark.parametrize(
         "candidate, version",
         [
-            (ssl.PROTOCOL_TLSv1, ssl.PROTOCOL_TLSv1),
-            ("PROTOCOL_TLSv1", ssl.PROTOCOL_TLSv1),
-            ("TLSv1", ssl.PROTOCOL_TLSv1),
+            *([
+                (getattr(ssl, "PROTOCOL_TLSv1", None), getattr(ssl, "PROTOCOL_TLSv1", None)),
+                ("PROTOCOL_TLSv1", getattr(ssl, "PROTOCOL_TLSv1", None)),
+                ("TLSv1", getattr(ssl, "PROTOCOL_TLSv1", None)),
+            ] if hasattr(ssl, "PROTOCOL_TLSv1") else []),
             (ssl.PROTOCOL_SSLv23, ssl.PROTOCOL_SSLv23),
         ],
     )