Some estimate his price-tag to be £80 million ($106 million), while others suggest it could be a Premier League record-breaking £100 million ($133 million).
Pogba was youth player with United five years ago, and played a single season with the first team squad. He left the club under a cloud when he did not sign a new contract after that season. Sir Alex Ferguson, the manager at the time, said that Pogba's move was "disappointing."
"I don't think he showed us any respect at all, to be honest," he added.
New United manager Jose Mourinho clearly has no such doubts as he looks to get his superstar team ready for the next season, with the recent announcement that Swedish striker Zlatan Ibrahimovic will join the club.
Pogba would undoubtedly make a great addition to United's midfield. After leaving the team in 2012 he joined the Italian side Juventus, where he has so far scored 28 goals.
The United announcement is unlikely to come before the end of Euro 2016, where Pogba is currently on duty with France. The team beat Iceland 5-2 in the quarter-finals on Sunday, with Pogba scoring the second goal. France plays Germany in the semi-final on Thursday.
United legend Ryan Giggs is fan of Pogba, commenting during the Iceland game that he is "a talented player. He can shoot with both feet, pass the ball with both feet and you don't realise how big he is. Physically he's immense, he's strong, he's quick, and he's got everything. He has got everything."