Reports linked the Ladbrokes Premiership leaders with a move for the Dundee custodian, who is currently on loan at Hibs.
With Craig Gordon out for 12 weeks through injury, the Parkhead side have made no secret about their desire to find an experienced back-up to Dorus de Vries, who started the 3-1 victory over Hearts.
Rodgers remains hopeful that they’ll manage to get a deal over the line, along with adding another outfield player to the squad.
He said: “I’ll need to speak to Peter [Lawwell] but there was nothing changed before the game. Ideally we’ll have a goalkeeper and one other.
“I could sit here all day and go through the names but he [Bain] is one of many mentioned. We want a goalkeeper that fits our idea of how we play. That’s important.”
On his side’s win, he added: “‘It was a very good performance both halves. We created chances and that’s the only disappointment from the evening – not to have scored more goals.”
While Hearts boss Craig Levein was pleased with the manner in which his side responded to going three goals down in 35 minutes at Parkhead, as Kyle Lafferty netted the only goal of the second half, he was left sweating over the fitness of Harry Cochrane after the 16-year-old was forced off with a neck injury, which required attention from the ambulance attendants.
He said: “It was his neck which is the worrying thing. We were late out for the second half because the doctors were all in the dressing room. We have to try and find out what’s happened.”