diff --git a/Lib/test/test_socket.py b/Lib/test/test_socket.py
index d277e36..ede1038 100644
--- a/Lib/test/test_socket.py
+++ b/Lib/test/test_socket.py
@@ -1168,6 +1168,12 @@
     @unittest.skipUnless(support.is_resource_enabled('network'),
                          'network is not enabled')
     def test_idna(self):
+        # Check for internet access before running test (issue #12804).
+        try:
+            socket.gethostbyname('python.org')
+        except socket.gaierror as e:
+            if e.errno == socket.EAI_NODATA:
+                self.skipTest('internet access required for this test')
         # these should all be successful
         socket.gethostbyname('испытание.python.org')
         socket.gethostbyname_ex('испытание.python.org')
diff --git a/Lib/test/test_urllib2net.py b/Lib/test/test_urllib2net.py
index 5fcb4cb..fc5527e 100644
--- a/Lib/test/test_urllib2net.py
+++ b/Lib/test/test_urllib2net.py
@@ -83,12 +83,13 @@
     def test_close(self):
         # calling .close() on urllib2's response objects should close the
         # underlying socket
-
-        response = _urlopen_with_retry("http://www.python.org/")
-        sock = response.fp
-        self.assertTrue(not sock.closed)
-        response.close()
-        self.assertTrue(sock.closed)
+        url = "http://www.python.org/"
+        with support.transient_internet(url):
+            response = _urlopen_with_retry(url)
+            sock = response.fp
+            self.assertTrue(not sock.closed)
+            response.close()
+            self.assertTrue(sock.closed)
 
 class OtherNetworkTests(unittest.TestCase):
     def setUp(self):