The 21-year-old defender, who had been on loan with the Clarets since August, is set to sign a three-and-a-half year contract at Turf Moor.
The deal is reward for the positive impression Keane has made with the Premier League newcomers, having established himself in the side since November.
Manager Sean Dyche said: "Michael is a good young player and we see a lot of development in him as he's on a good pathway at the moment and we're delighted that he has decided to join us."
Keane's signing will ease some of Dyche's defensive concerns with Kevin Long facing a long spell out and Mike Duff and Jason Shackell having also had time on the sidelines recently.
Keane came through the United youth set-up and signed professionally for the club in 2011.
He had loan spells at Leicester, Derby and Blackburn and then made two appearances for United at the start of this season before joining Burnley on August deadline day.
He has since made 11 appearances for Dyche's side.