diff --git a/asyncpg/connect_utils.py b/asyncpg/connect_utils.py
index 07c4fdde..70b7a6ba 100644
--- a/asyncpg/connect_utils.py
+++ b/asyncpg/connect_utils.py
@@ -934,6 +934,12 @@ def data_received(self, data: bytes) -> None:
             # sslmode=prefer. But be extra sure to disallow insecure
             # connections when the ssl context asks for real security.
             self.on_data.set_result(False)
+        elif data.startswith(b'E'):
+            message = data[1:].rstrip(b'\x00\r\n').decode(
+                'utf-8', errors='replace')
+            self.on_data.set_exception(
+                exceptions.InterfaceError(
+                    message or 'server error during SSL negotiation'))
         else:
             self.on_data.set_exception(
                 ConnectionError(
diff --git a/tests/test_connect.py b/tests/test_connect.py
index 955fb825..741d89f8 100644
--- a/tests/test_connect.py
+++ b/tests/test_connect.py
@@ -99,6 +99,19 @@ def mock_dev_null_home_dir():
         yield
 
 
+class TestTLSUpgradeProto(tb.TestCase):
+
+    async def test_error_response_preserves_server_message(self):
+        context = ssl.SSLContext(ssl.PROTOCOL_TLS_CLIENT)
+        proto = connect_utils.TLSUpgradeProto(
+            self.loop, 'localhost', 5432, context, False)
+
+        proto.data_received(b'Etoo many connections\n\x00')
+        with self.assertRaisesRegex(
+                exceptions.InterfaceError, 'too many connections'):
+            await proto.on_data
+
+
 class TestSettings(tb.ConnectedTestCase):
 
     async def test_get_settings_01(self):