As it stands, the 30-year-old's current deal is due to expire in the summer, which would make him a free agent.
However, despite suggesting that he would like to stay at Turf Moor, Grant has admitted that he would have to take a pay cut to commit his future to the Clarets.
"We're getting there. It's not a quick process by any means," he told the Lancashire Evening Telegraph. "It's been well documented the club are looking to tighten their belts slightly and players are having to take a drop and I'm no different in that sense.
"It's obviously something that has to be thought about a lot from my part as well, there's a lot of work going into it behind the scenes, but we're getting there."
Grant joined Burnley from Sheffield Wednesday in 2010.