The 18-year-old has scored 13 times and registered five assists in 43 appearances for Barca since breaking into the first team, including five goals and four assists in 10 outings this term.
The Spain international is expected to be out until March with a knee injury but his club are said to be working on a new contract until the summer of 2026.
According to Sport, super-agent Jorge Mendes set up a €150m (£134m) move to United over the summer, but Barca quickly rejected the 'dizzying' proposal.
The report claims that the La Liga giants were not even prepared to entertain the offer due to Fati's importance to their long-term project, particularly with Lionel Messi expected to leave next summer.
The teenager is believed to have a £367m release clause in his current deal at Camp Nou.