diff --git a/Lib/socket.py b/Lib/socket.py
--- a/Lib/socket.py
+++ b/Lib/socket.py
@@ -263,17 +263,17 @@ class socket(_socket.socket):
             self._check_sendfile_params(file, offset, count)
             sockno = self.fileno()
             try:
                 fileno = file.fileno()
             except (AttributeError, io.UnsupportedOperation) as err:
                 raise _GiveupOnSendfile(err)  # not a regular file
             try:
                 fsize = os.fstat(fileno).st_size
-            except OSError:
+            except OSError as err:
                 raise _GiveupOnSendfile(err)  # not a regular file
             if not fsize:
                 return 0  # empty file
             blocksize = fsize if not count else count
 
             timeout = self.gettimeout()
             if timeout == 0:
                 raise ValueError("non-blocking sockets are not supported")
diff --git a/Lib/test/test_socket.py b/Lib/test/test_socket.py
--- a/Lib/test/test_socket.py
+++ b/Lib/test/test_socket.py
@@ -1480,16 +1480,37 @@ class GeneralModuleTests(unittest.TestCa
         # type and populates the socket object.
         #
         # On Windows this trick won't work, so the test is skipped.
         fd, _ = tempfile.mkstemp()
         with socket.socket(family=42424, type=13331, fileno=fd) as s:
             self.assertEqual(s.family, 42424)
             self.assertEqual(s.type, 13331)
 
+    @unittest.skipUnless(hasattr(os, 'sendfile'), 'test needs os.sendfile()')
+    def test__sendfile_use_sendfile(self):
+        class File:
+            def __init__(self, fd):
+                self.fd = fd
+
+            def fileno(self):
+                return self.fd
+        tests = [
+            (os.open(os.curdir, os.O_RDONLY), socket._GiveupOnSendfile),
+            (2**1000, OverflowError),
+            (None, TypeError),
+        ]
+        for fd, exc in tests:
+            with self.subTest(fd=fd, exc=exc.__name__):
+                with socket.socket() as sock:
+                    if exc == socket._GiveupOnSendfile:
+                        os.close(fd)
+                    self.assertRaises(exc, sock._sendfile_use_sendfile, File(fd))
+
+
 @unittest.skipUnless(HAVE_SOCKET_CAN, 'SocketCan required for this test.')
 class BasicCANTest(unittest.TestCase):
 
     def testCrucialConstants(self):
         socket.AF_CAN
         socket.PF_CAN
         socket.CAN_RAW