Manager Steve Bruce is keen to bring in a trio of players from his former club, with fellow defender Wes Brown and midfielder Darron Gibson also set for the Old Trafford exit door following the Red Devils' captures of Phil Jones and Ashley Young.
The Republic of Ireland international, who can play anywhere across the back-line, has made almost 400 appearances for Sir Alex Ferguson's side, and is also reportedly interesting both Arsenal and Stoke City.
However, Sunderland remain the favourites to secure his signature, although Quinn revealed that they are currently having to play a waiting game following the birth of O'Shea's first child.
"We are in there trying hard for John and will see if something can unravel. We feel it is getting closer," Quinn said in 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.
"There is interest [from other clubs], naturally. All you can do is make your own pitch but we would love if John chose us."
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