The England Under-21 international becomes the third player to move to Turf Moor on transfer deadline day and will remain with the Clarets until January
The deal, subject to final Premier League approval, was completed late on deadline day and sees the England Under-21s international join until January.
Keane becomes the third player to move to Turf Moor in the final hours of the transfer window after George Boyd and Nathaniel Chalobah arrived from Hull City and Chelsea respectively earlier in the day.
Keane has enjoyed loan spells in the Championship with Leicester City and Derby County, as well as Burnley's local rivals Blackburn Rovers, but his upcoming spell with the Clarets is his first taste of Premier League football away from Old Trafford.
The 21-year-old made his league debut for the Red Devils during their 1-1 draw with Sunderland earlier in the season following an injury to Chris Smalling, but has been deemed surplus to requirements by Louis van Gaal as the Dutch coach looks for him to gain more experience away from Old Trafford.