The Black Cats reacted swiftly to claims that the 25-year-old Ghana international's name has been circulated to prospective purchasers.
In a statement, Quinn said: "The suggestion that we have spoken to an agent to sell this player on our behalf is embarrassing to all concerned.
"I hope our fans would be smart enough to realise that this is probably some other club or outside agent's way of attempting to destabilise our relationship with one of our players."
A report suggested Gyan was being put up for sale to help fund a summer spending spree.
His situation has been in the spotlight for several weeks amid reports that Tottenham could be ready to test Sunderland's resolve to hang on to him this summer.
The former Rennes striker headed for the Stadium of Light in a £13million-plus move from Rennes last summer and scored 11 goals in his first campaign on Wearside.
Manager Steve Bruce bought him to partner Darren Bent in attack, although his plans had to be ripped up when the England international left for Aston Villa in January.
Bruce's focus since has been the search for a player to complement the African and he has been linked with a long list of names, with Spurs pair Peter Crouch and Roman Pavlyuchenko prominent among them.
The noises about Gyan's future have refused to go away, but with loan signing Danny Welbeck having returned to Manchester United and Fraizer Campbell facing most of next season on the sidelines with another serious knee injury, the club's striking resources are all but exhausted.
If they did decide to cash in on a man on whom they could expect to make a profit, it would leave the manager with an entire attacking unit to rebuild and another major gap to plug following midfielder Jordan Henderson's departure for Liverpool.
Bruce is already heavily engaged in his search for reinforcements, with deals for Liverpool's David Ngog and South Korea international Ji Dong-won close to being finalised.
Elsewhere, he has been given permission to talk to Manchester United trio Darron Gibson, Wes Brown and John O'Shea after lodging a joint bid totalling more than £12million.
The Black Cats have been encouraged by the reaction from the Gibson camp, but Brown is understood to be considering his options while O'Shea will perhaps take the most persuading to leave Old Trafford.
Sunderland have also registered an interest in Birmingham midfielder Craig Gardner, who is certain to attract interest from other Premier League clubs, although derby rivals Newcastle insist they will not be among them.
Bruce also hopes to push through Bosman free transfer deals for Coventry keeper Keiren Westwood and Birmingham midfielder Sebastian Larsson after wasting no time in making his interest known.
Quinn said: "What I can say is that we are in a number of meaningful negotiations to bring players in and when there's anything definitive to report, it will be on our club website for our fans to read."