Celtic boss Brendan Rodgers admits the door is still open.
Charlie Mulgrew also remains hopeful that Greg Taylor will stay at Parkhead despite the return of the prodigal son Kieran Tierney.
Taylor, who was linked with a move to Greek side PAOK earlier this week, will become a free agent in a matter of days after failing to sign a new deal with the Scottish champions
Rodgers is refusing to throw the towel in, whilst Mulgrew reckons there could still be a change of heart by all parties at the 11th hour.
There is a school of thought that says Taylor wants to play regular first-team football and not play second fiddle to Tierney.
However, as Mulgrew was at pains to point out, Celtic played 57 games in all competitions last season, and with Tierney's track record of injuries at Arsenal, his game time will need to be managed.
That means there would still be ample minutes for Taylor even if he is viewed as the backup left back next season.
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Mulgrew said: "As a Celtic fan, I would love to see Greg Taylor stick around. I think he does that inverted role brilliantly, and between him and Kieran Tierney, they can still have enough games between them over a season.
“It's unrealistic to play every single game at Celtic because we are approaching 60 games in a season, so I think there's plenty there for the two of them.
“You imagine KT will be looking to stay fit, but you don't know how realistic that is. He's probably not going to play 60 games at the level, so I'd love to see Taylor stay because I think he offers a lot for the changing room.
“Tierney will still need that competition to know that he's not just going to be there every game. Taylor would provide that perfectly, and then he's got the determination and the attitude to be in the Celtic team.
"He’ll want to fight to be in there as well, and you just never know, Taylor could get the jersey. You never know, football is a funny game sometimes."