diff --git a/asyncpg/connect_utils.py b/asyncpg/connect_utils.py
index f98935f5..c09bf5e0 100644
--- a/asyncpg/connect_utils.py
+++ b/asyncpg/connect_utils.py
@@ -757,10 +757,9 @@ async def _connect_addr(
 
     params_input = params
     if callable(params.password):
-        if inspect.iscoroutinefunction(params.password):
-            password = await params.password()
-        else:
-            password = params.password()
+        password = params.password()
+        if inspect.isawaitable(password):
+            password = await password
 
         params = params._replace(password=password)
     args = (addr, loop, config, connection_class, record_class, params_input)
diff --git a/tests/test_connect.py b/tests/test_connect.py
index b78f4f48..34ffbb34 100644
--- a/tests/test_connect.py
+++ b/tests/test_connect.py
@@ -282,6 +282,25 @@ async def get_wrongpassword():
                 user='password_user',
                 password=get_wrongpassword)
 
+    async def test_auth_password_cleartext_callable_awaitable(self):
+        async def get_correctpassword():
+            return 'correctpassword'
+
+        async def get_wrongpassword():
+            return 'wrongpassword'
+
+        conn = await self.connect(
+            user='password_user',
+            password=lambda: get_correctpassword())
+        await conn.close()
+
+        with self.assertRaisesRegex(
+                asyncpg.InvalidPasswordError,
+                'password authentication failed for user "password_user"'):
+            await self._try_connect(
+                user='password_user',
+                password=lambda: get_wrongpassword())
+
     async def test_auth_password_md5(self):
         conn = await self.connect(
             user='md5_user', password='correctpassword')