The Premier League giants have been monitoring progress of the Brazilian International for over the last campaign and have maintained close contacts over a potential move this summer.
Fabinho is represented by the high-profile agent Jorge Mendes, who also represents Red Devils boss Jose Mourinho and Spanish goalkeeper David De Gea.
Monaco signed the Brazilian on a permanently transfer only last summer after a successful two-season loan spell from Real Madrid prior to that.
The 22-year-old failed to appear in only four league matches for Monaco last season, netting six goals and providing assists for another four goals along the way.
Napoli and Manchester City have also reportedly registered interest to sign the former Real Madrid Castilla player, while a proposed £12 million move to Naples fell through earlier this summer.
Fabinho has three years left on his contract with the Ligue 1 outfit, who are poised to set an asking price in excess of £20 million, should Man Utd come calling for the highly-rated full back.
Jose Mourinho is said to be unconvinced by his available right backs at Man Utd and is keen to make defensive reinforcements this summer. The former Chelsea boss is unsure by the attributes offered by Italian Matteo Darmian and Uruguayan Guillermo Varela, who currently remain the only out and out right full backs at Man Utd for next season.