diff --git a/pymongo/asynchronous/mongo_client.py b/pymongo/asynchronous/mongo_client.py
index 9dba97d12..f7fc8e5e8 100644
--- a/pymongo/asynchronous/mongo_client.py
+++ b/pymongo/asynchronous/mongo_client.py
@@ -1185,7 +1185,7 @@ class AsyncMongoClient(common.BaseObject, Generic[_DocumentType]):
     def __del__(self) -> None:
         """Check that this AsyncMongoClient has been closed and issue a warning if not."""
         try:
-            if not self._closed:
+            if self._opened and not self._closed:
                 warnings.warn(
                     (
                         f"Unclosed {type(self).__name__} opened at:\n{self._topology_settings._stack}"
diff --git a/pymongo/synchronous/mongo_client.py b/pymongo/synchronous/mongo_client.py
index 21fa57b5d..5786bbf5a 100644
--- a/pymongo/synchronous/mongo_client.py
+++ b/pymongo/synchronous/mongo_client.py
@@ -1185,7 +1185,7 @@ class MongoClient(common.BaseObject, Generic[_DocumentType]):
     def __del__(self) -> None:
         """Check that this MongoClient has been closed and issue a warning if not."""
         try:
-            if not self._closed:
+            if self._opened and not self._closed:
                 warnings.warn(
                     (
                         f"Unclosed {type(self).__name__} opened at:\n{self._topology_settings._stack}"
diff --git a/test/asynchronous/test_auth.py b/test/asynchronous/test_auth.py
index e516ff679..06f7fb9ca 100644
--- a/test/asynchronous/test_auth.py
+++ b/test/asynchronous/test_auth.py
@@ -169,7 +169,7 @@ class TestGSSAPI(unittest.IsolatedAsyncioTestCase):
         client = AsyncMongoClient(mech_uri)
         await client[GSSAPI_DB].collection.find_one()
 
-        set_name = await client.admin.command(HelloCompat.LEGACY_CMD).get("setName")
+        set_name = async_client_context.replica_set_name
         if set_name:
             if not self.service_realm_required:
                 # Without authMechanismProperties
@@ -242,7 +242,7 @@ class TestGSSAPI(unittest.IsolatedAsyncioTestCase):
             thread.join()
             self.assertTrue(thread.success)
 
-        set_name = await client.admin.command(HelloCompat.LEGACY_CMD).get("setName")
+        set_name = async_client_context.replica_set_name
         if set_name:
             client = AsyncMongoClient(
                 GSSAPI_HOST,
@@ -296,7 +296,7 @@ class TestSASLPlain(unittest.IsolatedAsyncioTestCase):
         client = AsyncMongoClient(uri)
         await client.ldap.test.find_one()
 
-        set_name = await client.admin.command(HelloCompat.LEGACY_CMD).get("setName")
+        set_name = async_client_context.replica_set_name
         if set_name:
             client = AsyncMongoClient(
                 SASL_HOST,
diff --git a/test/test_auth.py b/test/test_auth.py
index 0bf0cfd80..fa3d0905b 100644
--- a/test/test_auth.py
+++ b/test/test_auth.py
@@ -169,7 +169,7 @@ class TestGSSAPI(unittest.TestCase):
         client = MongoClient(mech_uri)
         client[GSSAPI_DB].collection.find_one()
 
-        set_name = client.admin.command(HelloCompat.LEGACY_CMD).get("setName")
+        set_name = client_context.replica_set_name
         if set_name:
             if not self.service_realm_required:
                 # Without authMechanismProperties
@@ -242,7 +242,7 @@ class TestGSSAPI(unittest.TestCase):
             thread.join()
             self.assertTrue(thread.success)
 
-        set_name = client.admin.command(HelloCompat.LEGACY_CMD).get("setName")
+        set_name = client_context.replica_set_name
         if set_name:
             client = MongoClient(
                 GSSAPI_HOST,
@@ -296,7 +296,7 @@ class TestSASLPlain(unittest.TestCase):
         client = MongoClient(uri)
         client.ldap.test.find_one()
 
-        set_name = client.admin.command(HelloCompat.LEGACY_CMD).get("setName")
+        set_name = client_context.replica_set_name
         if set_name:
             client = MongoClient(
                 SASL_HOST,