Grant, who is out of contract at the end of the season, has been offered a new deal by the Championship club.
A number of clubs are thought to be monitoring Grant's situation at Turf Moor, with the former Derby and Sheffield Wednesday an attractive option on a possible free transfer in the summer.
Grant has been an ever-present for Burnley this season and Clarets boss Sean Dyche is keen to keep hold of the 30-year-old.
The player's agent, Darryl Powell, has confirmed Grant is considering the contract offer from Burnley and that the club are not rushing him into making a decision on his future.
"Lee is is considering the contract offer made to him by Burnley," Powell told Sky Sports.
"He has a lot of things to consider and his club are allowing him that time."