Bellamy, who spent a successful six-month loan spell at the Hoops in 2005, had been wanted by manager Neil Lennon to play up front.
But the 32-year-old told his former team-mate he has no intention of returning to Glasgow on loan and he may now try to revive his career at Barclays Premier League giants Manchester City.
Lennon has now turned his attention to other players to bolster his frontline – and Torino hitman Bianchi, above left, is on his radar.
The 28-year-old former Manchester City player is set to leave his Serie B club and Celtic have been in contact with his agent, his brother Riccardo, who said: "I can confirm that Celtic are interested, but they are not the only club."
Meanwhile, Lennon could make another bid to lure Fraser Forster back to Parkhead if the player doesn't start the English season as first-choice keeper at Newcastle United.
Forster made a blunder in Newcastle's pre-season defeat to Leeds United at the weekend and that could affect his chances of starting the new campaign between the sticks for Alan Pardew's Premiership stars.
Pardew has said he will play Steve Harper and Tim Krul in his side's next two pre-season matches before deciding who to play in his first team.
He said: "Fraser is not out of the frame in any shape or form. The plan is to play Harper at Gateshead and then Krul against Fiorentina."
However, if Pardew decides to leave out Forster, who spent a successful nine months on loan in Scotland last season, then Lennon intends to make a move.
The Irishman has already had two cash bids for the 6ft 7in stopper turned down and his offer to take him back to Parkhead on loan was also rejected.
Lennon knows Forster is keen to play regular first-team football and would be unhappy to warm the bench at St James's Park.