diff --git a/acme/acme/client.py b/acme/acme/client.py
index 9455159de..fa903f0e6 100644
--- a/acme/acme/client.py
+++ b/acme/acme/client.py
@@ -519,7 +519,12 @@ class ClientNetwork(object):  # pylint: disable=too-many-instance-attributes
         self._default_timeout = timeout
 
     def __del__(self):
-        self.session.close()
+        # Try to close the session, but don't show exceptions to the
+        # user if the call to close() fails. See #4840.
+        try:
+            self.session.close()
+        except Exception:  # pylint: disable=broad-except
+            pass
 
     def _wrap_in_jws(self, obj, nonce):
         """Wrap `JSONDeSerializable` object in JWS.
diff --git a/acme/acme/client_test.py b/acme/acme/client_test.py
index cd1a90645..54652b46c 100644
--- a/acme/acme/client_test.py
+++ b/acme/acme/client_test.py
@@ -600,12 +600,19 @@ class ClientNetworkTest(unittest.TestCase):
             mock.ANY, mock.ANY, verify=mock.ANY, headers=mock.ANY,
             timeout=45)
 
-    def test_del(self):
+    def test_del(self, close_exception=None):
         sess = mock.MagicMock()
+
+        if close_exception is not None:
+            sess.close.side_effect = close_exception
+
         self.net.session = sess
         del self.net
         sess.close.assert_called_once_with()
 
+    def test_del_error(self):
+        self.test_del(ReferenceError)
+
     @mock.patch('acme.client.requests')
     def test_requests_error_passthrough(self, mock_requests):
         mock_requests.exceptions = requests.exceptions