The 30-year-old, who has been one of his side's key players this season, will be out of contract when his current deal expires at the end of the season.
He has been offered fresh terms, but his agent Darryl Powell has suggested that Grant has a number of other avenues to explore.
"Lee is considering the contract offer made to him by Burnley. He has a lot of things to consider and his club are allowing him that time," Powell told Sky Sports News.
Derby County are reported to be leading the race for his signature.