diff --git a/.evergreen/generated_configs/variants.yml b/.evergreen/generated_configs/variants.yml
index a05cc61f05..33b8e0ba02 100644
--- a/.evergreen/generated_configs/variants.yml
+++ b/.evergreen/generated_configs/variants.yml
@@ -332,7 +332,7 @@ buildvariants:
   # Green framework tests
   - name: green-eventlet-rhel8
     tasks:
-      - name: .test-standard .standalone-noauth-nossl .python-3.9 .sync
+      - name: .test-standard .python-3.9 .sync
     display_name: Green Eventlet RHEL8
     run_on:
       - rhel87-small
@@ -340,7 +340,7 @@ buildvariants:
       GREEN_FRAMEWORK: eventlet
   - name: green-gevent-rhel8
     tasks:
-      - name: .test-standard .standalone-noauth-nossl .sync
+      - name: .test-standard .sync
     display_name: Green Gevent RHEL8
     run_on:
       - rhel87-small
diff --git a/.evergreen/scripts/generate_config.py b/.evergreen/scripts/generate_config.py
index 3a386be4f7..f76c0d1e04 100644
--- a/.evergreen/scripts/generate_config.py
+++ b/.evergreen/scripts/generate_config.py
@@ -301,11 +301,11 @@ def create_green_framework_variants():
     variants = []
     host = DEFAULT_HOST
     for framework in ["eventlet", "gevent"]:
-        tasks = [".test-standard .standalone-noauth-nossl .sync"]
+        tasks = [".test-standard .sync"]
         if framework == "eventlet":
             # Eventlet has issues with dnspython > 2.0 and newer versions of CPython
             # https://jira.mongodb.org/browse/PYTHON-5284
-            tasks = [".test-standard .standalone-noauth-nossl .python-3.9 .sync"]
+            tasks = [".test-standard .python-3.9 .sync"]
         expansions = dict(GREEN_FRAMEWORK=framework)
         display_name = get_variant_name(f"Green {framework.capitalize()}", host)
         variant = create_variant(tasks, display_name, host=host, expansions=expansions)
diff --git a/test/asynchronous/test_monitor.py b/test/asynchronous/test_monitor.py
index 55a20d7643..dde8976c06 100644
--- a/test/asynchronous/test_monitor.py
+++ b/test/asynchronous/test_monitor.py
@@ -28,7 +28,7 @@
 from test.asynchronous.utils import (
     async_wait_until,
 )
-from test.utils_shared import ServerAndTopologyEventListener
+from test.utils_shared import ServerAndTopologyEventListener, gevent_monkey_patched
 
 from pymongo.periodic_executor import _EXECUTORS
 
@@ -58,6 +58,9 @@ async def create_client(self):
         return client
 
     @unittest.skipIf("PyPy" in sys.version, "PYTHON-5283 fails often on PyPy")
+    @unittest.skipIf(
+        gevent_monkey_patched(), "PYTHON-5516 Resources are not cleared when using gevent"
+    )
     async def test_cleanup_executors_on_client_del(self):
         with warnings.catch_warnings(record=True) as w:
             warnings.simplefilter("always")
diff --git a/test/test_monitor.py b/test/test_monitor.py
index 8bcdf7130a..c10662c893 100644
--- a/test/test_monitor.py
+++ b/test/test_monitor.py
@@ -28,7 +28,7 @@
 from test.utils import (
     wait_until,
 )
-from test.utils_shared import ServerAndTopologyEventListener
+from test.utils_shared import ServerAndTopologyEventListener, gevent_monkey_patched
 
 from pymongo.periodic_executor import _EXECUTORS
 
@@ -58,6 +58,9 @@ def create_client(self):
         return client
 
     @unittest.skipIf("PyPy" in sys.version, "PYTHON-5283 fails often on PyPy")
+    @unittest.skipIf(
+        gevent_monkey_patched(), "PYTHON-5516 Resources are not cleared when using gevent"
+    )
     def test_cleanup_executors_on_client_del(self):
         with warnings.catch_warnings(record=True) as w:
             warnings.simplefilter("always")