The 24-year-old came up through the ranks of the Red Devils' youth system. After failing to cement a place in the first team, the centre-back was allowed to leave Old Trafford by their former manager Louis van Gaal. He joined Burnley on a permanent transfer in January 2015.
Keane has been in fine form for Sean Dyche's side this season and his form has seen him attract interest from the 20-time English champions, Everton and Liverpool. According to The Times, United and Burnley are "very close" to agreeing a deal that is believed to be about £25m (32.4m).
United have managed to beat the competition from the two Merseyside clubs in securing the England international's services. He is set to become Jose Mourinho's first signing of the summer transfer window.
The 20-time English champions allowed Keane to join Burnley for £2m ($2.6m) and inserted a 25% sell-on clause. This means that United will have to pay far less than other clubs in signing him.
Dyche has already admitted that Burnley would be open to the idea of sanctioning Keane's sale "if all parties agree".
"It is more of case of doing what is right in any situation," Dyche told talkSPORT.
"If all parties agree, that might be something the club have to look at just by the realities of what it is in the Premier League but we are certainly in very healthy shape so we don't have to make any decisions for any other reason than we think it is appropriate."
"It is the balance of what you are doing at a club like this," he said. "You want your players to be successful, which is good for them, good for the team, good for the club. On the other hand, that does bring interest."
With Keane's imminent return to United, it will be the second successive summer transfer window when the Red Devils have re-signed a former player. Last summer, Mourinho brought Paul Pogba back to Old Trafford from Juventus on a world-record £89m fee.