The Scottish stopper joined Brendan Rodgers side on loan from Dundee in January and has impressed in recent weeks.
Stoke will try and sign the 26-year-old when his contract expires at the end of the season, with the future of Jack Butland uncertain amid links with Arsenal and Liverpool.
However, Wolves, Hearts, Hibernian and Hearts are also reportedly keen to sign Bain, which could lead to a bidding war this summer.
The 26-year-old recently spoke about staying at Parkhead beyond the end of the season and practically issued a come-and-get-me-plea to the Hoops.
He said via The Scottish Sun
“I would love to be here. I’ve loved every minute of being at this club and the fans have been fantastic with me.”
“I would like it to be here as this club has got everything for me to improve, I’d love to stay here and if I continue to play well then it stands me in good stead.”
Personally, Bain would be a good option for Celtic, but the Premier League interested could see them miss out.
Despite only making two appearances for the Scottish champions this season, he’s shown glimpses he could replace first-choice Craig Gordon, as the 35-year-old can’t be relied upon as a long-term option.