Wearsiders boss Steve Bruce has already made significant additions to his squad this summer, with Seb Larsson, Keiren Westwood, Connor Wickham, Dong-Won Ji and Craig Gardner already arriving at the Stadium of Light.
But Bruce is keen to add some valuable experience to his side and feels that 30-year-old O'Shea - who has won 15 trophies in 13 years at Old Trafford - is a player who can bring plenty of it.
"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. [So] 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."
Stoke and Arsenal have also been linked with a move for O'Shea and it appears the Irishman's United career is coming to end, while his team-mates Gibson and Wes Brown also look likely to be heading out of the Old Trafford exit door.
Sunderland are believed to have bid for all three players, but Quinn revealed that negotiations with Gibson are moving slowly.
"That one is a little bit further away," Quinn said. "We would certainly still like it to happen but we will have to see."