diff --git a/pymongo/asynchronous/mongo_client.py b/pymongo/asynchronous/mongo_client.py
index e347e1de56..a0f1397a4d 100644
--- a/pymongo/asynchronous/mongo_client.py
+++ b/pymongo/asynchronous/mongo_client.py
@@ -2752,8 +2752,10 @@ async def __aexit__(
         try:
             await self.handle(exc_type, exc_val)
         finally:
-            if self._pool_checkout is not None:
-                await self._pool_checkout.__aexit__(exc_type, exc_val, exc_tb)
+            pool_checkout = self._pool_checkout
+            self._pool_checkout = None  # to break the reference cycle before cleanup can raise
+            if pool_checkout is not None:
+                await pool_checkout.__aexit__(exc_type, exc_val, exc_tb)
 
     @classmethod
     def for_existing_conn(
diff --git a/pymongo/synchronous/mongo_client.py b/pymongo/synchronous/mongo_client.py
index 71ba6dd181..574fbc00de 100644
--- a/pymongo/synchronous/mongo_client.py
+++ b/pymongo/synchronous/mongo_client.py
@@ -2741,8 +2741,10 @@ def __exit__(
         try:
             self.handle(exc_type, exc_val)
         finally:
-            if self._pool_checkout is not None:
-                self._pool_checkout.__exit__(exc_type, exc_val, exc_tb)
+            pool_checkout = self._pool_checkout
+            self._pool_checkout = None  # to break the reference cycle before cleanup can raise
+            if pool_checkout is not None:
+                pool_checkout.__exit__(exc_type, exc_val, exc_tb)
 
     @classmethod
     def for_existing_conn(