The Black Cats boss Steve Bruce made a triple bid for United trio of O'Shea, Darron Gibson and Wes Brown earlier in the summer and they appear close to landing the defender.
Chairman Quinn is optimistic he can lure his fellow Irishman to the Stadium of Light.
"We are in there trying hard for John and will see if something can unravel. We feel it is getting closer," Quinn said.
"John had good family news in the last few days with the birth of his child. We've had to back off which is only correct. But the players are back in and we will see if something can unravel."