The 26-year-old was originally signed as an emergency back-up on a loan deal from Dundee in January after an injury to first-choice goalkeeper Craig Gordon.
Having been impressed in the 3-2 win over Rangers at Ibrox on Sunday, manager Brendan Rodgers is now looking to sign the stopper on a long-term basis.
The Ladbrokes Premiership leaders will have competition from Hibs and Hearts for his signature.
Neil Lennon, who originally signed the player in the January window, only to agree to cut the deal short in order to secure the services of Scott Allan, is still considering a summer move.
Hearts, meanwhile, will likely be looking for a new No.1 with current keeper Jon McLaughlin expected to leave when his one-year deal expires. Bain grew up a Hearts supporter.