Inter Milan have confirmed their interest in Cardiff's Gary Medel but have given up on Man City forward Stevan Jovetic.
Jeffrey Schlupp and Liam Moore have signed new deals with Premier League newcomers Leicester City.
Grant, 30, out of contract at Turf Moor, has agreed a three-year deal with the Rams, according to BBC Radio Derby.
The player spent seven years at Derby after progressing through their academy and joined Burnley from Sheffield Wednesday in July 2010.
"The process of bringing players in started a few weeks ago once we assured our safety in the Championship and we're quite a way down the line with one or two targets already," Derby boss Nigel Clough told the club's official website.
"We should have some news this week if things work out how we want them to.
"There's a few more on our list that we're working hard on and remain hopeful about bringing in and we'll put plenty of hard work in to get those over the line if we can."