The Wales international is now free to speak to other clubs, with his existing deal with the Hoops due to expire at the end of the season.
However, Celtic manager Martin O’Neill and Hartson’s agent Jonathan Barnett were encouraged by the latest round of talks during the week and Quinn now expects a quick conclusion.
“I don’t think that a move will happen,” said Quinn.
“I’m pretty confident that John Hartson will be staying at Celtic, not just this season but for one or two more seasons to come.
“Negotiations are very well advanced and they are advanced in a direction that makes me confident that those Celtic fans nervous about John Hartson can relax.”
Quinn is also optimistic of retaining the services of highly-rated manager O’Neill and believes there are enough challenges lying ahead to keep the Irishman in Glasgow.
Meanwhile, Celtic star Bobo Balde has been offered a new deal but he has yet to decide whether he wants to extend his stay at Parkhead.