The stopper who set a new shut-out record last season while on his second loan spell at Parkhead is Neil Lennon's first choice.
But if the Newcastle United keeper refuses the four-year deal on the table – believed to be worth around £12,000 per week – the Hoops will move on to the other names they already have on their shortlist.
Forster's agent, Mark Curtis, has held talks with Lennon and chief executive Peter Lawwell.
But with the transfer window now open, and Lennon recognising the need to get the position filled before other clubs begin to make their moves in the market, the SPL champions are eager to hear from Forster, who has just returned from holiday.
While he could earn more if he joined one of the clubs in England who are showing an interest in him, Celtic believe their trump card is that he is already established as the No.1 at Parkhead, and would retain that position.
Craig Gordon is one keeper who could find himself moving up the small list of candidates if Forster can't reach agreement with the Hoops, while former Hoops hero, Artur Boruc – who retains a big ally in Lennon – has also been linked with a return to the club he left to join Fiorentina two years ago.
Gordon – whose last move was a £9million transfer to Sunderland – is waiting to see who will offer him a fresh start following his release by the Black Cats.
Former Sunderland boss Steve Bruce, now in charge of Hull City, has already added his name to the list of men who would like to speak to Gordon about his future.
But the 29-year-old – who has been beset by injury to an arm and a knee – is keeping his cards close to his chest.
Gordon said: "I'm quite relaxed it will all get sorted out and I can get some first-team football under my belt."
If fan power has anything to with it, Boruc would find himself pulled back to Parkhead on a wave of emotion.
He retains strong feelings for Parkhead – where he played for four years under Gordon Strachan – but is not convinced a return to the SPL is what he needs at this stage of his career.
Boruc said: "When I think of Celtic, I think of warmth and love. But at this moment, I want to play in a better league and aim for the highest target."