diff --git a/tests/test_fetcher_ng.py b/tests/test_fetcher_ng.py
index 37eb67dd..99fcf258 100644
--- a/tests/test_fetcher_ng.py
+++ b/tests/test_fetcher_ng.py
@@ -13,10 +13,13 @@
 import unittest
 import tempfile
 import math
+import urllib3.exceptions
+import requests
 
 from tests import utils
 from tuf import exceptions, unittest_toolbox
 from tuf.ngclient._internal.requests_fetcher import RequestsFetcher
+from unittest.mock import patch
 
 logger = logging.getLogger(__name__)
 
@@ -121,6 +124,13 @@ def test_read_timeout(self):
 
         slow_server_process_handler.clean()
 
+    # Read/connect session timeout error
+    @patch.object(requests.Session, 'get', side_effect=urllib3.exceptions.TimeoutError)
+    def test_session_get_timeout(self, mock_session_get):
+        with self.assertRaises(exceptions.SlowRetrievalError):
+            self.fetcher.fetch(self.url)
+        mock_session_get.assert_called_once()
+
     # Simple bytes download
     def test_download_bytes(self):
         data = self.fetcher.download_bytes(self.url, self.file_length)
diff --git a/tuf/ngclient/_internal/requests_fetcher.py b/tuf/ngclient/_internal/requests_fetcher.py
index 3647e58f..60f293e8 100644
--- a/tuf/ngclient/_internal/requests_fetcher.py
+++ b/tuf/ngclient/_internal/requests_fetcher.py
@@ -77,7 +77,13 @@ def fetch(self, url: str) -> Iterator[bytes]:
         # requests as:
         #  - connect timeout (max delay before first byte is received)
         #  - read (gap) timeout (max delay between bytes received)
-        response = session.get(url, stream=True, timeout=self.socket_timeout)
+        try:
+            response = session.get(
+                url, stream=True, timeout=self.socket_timeout
+            )
+        except urllib3.exceptions.TimeoutError as e:
+            raise exceptions.SlowRetrievalError from e
+
         # Check response status.
         try:
             response.raise_for_status()