Both players have suffered injury setbacks as the Black Cats prepare to host Rotherham United at the Stadium of Light on Tuesday evening.
It is a huge game for both clubs, with Sunderland aiming to wrap up a play-off spot while victory for Rotherham will put them on the cusp of automatic promotion heading into the final game of the season on Saturday.
Winchester and Cirkin are unlikely to play a part against the Millers. The pair have made a combined 83 appearances for Sunderland this term.
Cirkin missed the Black Cats' emphatic 5-1 win over Cambridge United on Saturday after picking up an injury in training while Winchester was forced off with a groin problem in the first half of the game.
“We lost Dennis, which is really frustrating,” said Sunderland boss Alex Neil.
“We know that we’ve got gaps in the squad where if we lose a player, we are going to have those deficiencies exposed. Dennis is certainly one of those players.
“It's an injury issue with Dennis. In terms of his return, providing we extend the season (get into the play-offs), then I'd be very hopeful that Dennis would play a part.”
On Winchester, Neil added: “What I would say about Winny is that he's a robust lad, a hardy boy, so he's not one to come off if there's any chance you can continue."