The Brazil worldwide is linked with a move to United, with champions Manchester City having nearly landed the 25-year-old in January.
United lost to Chelsea in the FA Cup final at Wembley last weekend and finished second in the Premier League behind Manchester City.
Meanwhile, Barcelona have given a nod to Arsenal's pursuit of forward, Ousmane Dembele and the French worldwide might be seen moving to the Premier League next season.
Fred, who has five years left on his contract with Shakhtar, is likely to cost United around £52 million.
Although there are reports of Paul Pogba's departure in the summer but that is unlikely to happen and with Fred joining, the United midfield would be pretty powerful.
Pogba has endured a hard second season at Old Trafford and tension between himself and Mourinho has resulted in the midfielder being dropped from regular first team starter. He stated that both Manchester United and Manchester City have been interested in him for a while, but that the Red Devils have made the most significant progress.
Sources have told ESPN FC that United are keen to secure the services of the 25-year-old if they can agree a deal with Shakhtar Donetsk. That is, I can somewhat feel that Fred will be more expensive than Fernandinho and Teixeira. But I am focused exclusively on the World Cup. They need to reinforce their squad this summer to close the gap on champions Manchester City, who cantered to a league title with a 19-point lead by the end of the campaign.
In an interview with Shakhtar's official website, president Akhmetov confirmed negotiations took place over a deal worth around €50m (£43.7m).