Reports had suggested the Black Cats had circulated an email to alert potential suitors to the 25-year-old Ghanaian's availability. But Quinn has moved quickly to dismiss the speculation.
"The suggestion that we have spoken to an agent to sell this player on our behalf is embarrassing to all concerned," Quinn said in a statement.
"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."
Gyan scored 10 Premier League goals in his first season in the North East as the Stadium of Light side finished in tenth place.
The striker has this summer been linked with a move to Tottenham Hotspur, with Spurs boss Harry Redknapp believed to be keen on rebulding his front line.
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