The Black Cats have made a triple bid for O'Shea and United team-mates Darron Gibson and Wes Brown as they look to bolster their squad.
And United are ready to let the trio go on their way after losing their places in the United side.
Stoke may or may not be interested in a couple of those players, depending which reports you believe, but it is known that they are interested in O’Shea. But Quinn is optimistic he can entice the player 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 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."
"There is interest, naturally. All you can do is make your own pitch but we would love if John chose us," added Quinn.