The Guardian report that Burnley have paid the Red Devils a £2 million fee to acquire the 21-year-old's services on a permanent basis.
The deal is a reward for Keane's fine form at Turf Moor during the first half of the season. The 6ft 2in centre-back, who made a total of five appearances for Manchester United following his senior debut in 2011, impressed for the Clarets after establishing himself as a first-team regular in November.
Dyche delighted with Keane buy
“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," Burnley manager Sean Dyche told the club's official website. "We’re delighted that he has decided to join us.”
Keane, the twin brother of Manchester United striker Will, was sent on loan to Leicester City, Derby County and Blackburn Rovers before his spell with Burnley this season.
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Burnley favourites for relegation
And although Burnley are amongst the favourites to suffer from the Premier League this season - Dyche's side currently sit 19th in the table, having failed to win any of their four previous league fixtures - Keane clearly believes his development is best served at Turf Moor due to his exposure to regular first-team football.
Ironically, Keane may have made a similar number of first-team appearances this season had he opted to stay put at Old Trafford.
Paddy McNair has made nine Premier League appearances so far this term, while Tyler Blackett has featured in eight league games. Neither defender was rated as highly as Keane, who won United's Reserve Player of the Year award in 2012.
United set to strengthen defence
But with Manchester United poised to strengthen their defensive options - potentially with a big-name arrival, such as Borussia Dortmund captain Mats Hummels - Keane may have recognised that his chances of establishing himself in the Red Devils' first team were slim.
Phil Jones, Chris Smalling and Jonny Evans are Van Gaal's current first-choice defenders, but all have suffered various injury problems and dips in form this season.
Subsequently, the veteran Dutch coach will attempt to sign at least one top-level central defender either this month of when the transfer window re-opens in the summer.