The 26-year-old, who is on loan with the Scottish Premiership leaders, sees his contract with Neil McCann's side expire in the summer.
Bain joined the Parkhead side on deadline day in January and made his senior debut in the thrilling 3-2 Old Firm derby win over Rangers at Ibrox.
"I would love to be here," Bain told Sky Sports News ahead of Saturday's clash with Ross County.
"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.
"I think I've improved and my performances have been quite high considering I hadn't played games in so long, so it's only going to get better.
"I think being here has definitely given me an opportunity to be thought about for the Scotland squad. But it's not something that's on my mind all the time, I'm just enjoying being here."
Edinburgh rivals Hearts and Hibs are also believed to be vying for Bain's signature.