Grant, 30, has been an ever present for the Clarets this season, keeping 11 clean sheets in 41 games, and his current deal expires in the summer.
Burnley are actively seeking to reduce their wage bill ahead of next season, the last in which they will benefit from Barclays Premier League parachute payments.
Winger Ross Wallace has already accepted reduced terms to stay on, but the policy has so far prevented former Sheffield Wednesday man Grant from putting pen to paper.
He told the Lancashire Telegraph: "We're getting there. It's not a quick process by any means.
"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."
- Sports & Recreation