The 26-year-old goalkeeper arrived on loan from Dundee near the end of the most recent transfer deadline day after a temporary move to Hibernian had been cancelled and will become a free agent at the end of the campaign.
Having recently taken over from Dorus de Vries, who had deputised for injured number one Craig Gordon, Bain is hoping to impress boss Brendan Rodgers enough to win a new deal.
He said: “I would love to be here. I’ve loved every minute of being at this club and the fans have been fantastic with me.
“Here I can reach my potential with the staff and the players you train with every day.
“Because of the age I am it’s going to be a big move for me in terms of my career.
“I would like it to be here as this club has got everything for me to improve.”