The Black Cats have made a triple bid for O’Shea and his United team-mates Darron Gibson and Wes Brown.
“We are in there trying hard for John and will see if something can unravel. We feel it is getting closer,” Quinn told the Irish Daily Mirror.
“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.”
Arsenal and Stoke are also reportedly interested in the Republic of Ireland international.
“There is interest, naturally. All you can do is make your own pitch but we would love if John chose us,” added Quinn.
Quinn admits they are still working on luring Gibson to Sunderland after holding talks with the midfielder.
He continued: “That one is a little bit further away. We would certainly still like it to happen but we will have to see.”
