The French midfielder has attracted interest from a number of clubs across Europe following his impressive campaign with Monaco. The 22-year-old played 43 games in the defensive midfield position and contributed with three goals.
According to the Manchester Evening News (MEN), Bakayoko is yet to agree a deal with Chelsea and is keen on a move to Old Trafford in the summer. His interest in joining United is said to have peaked after they qualified for the Champions League by winning the Europa League.
Chelsea looked to be favourites to sign the France international after reports suggested that Antonio Conte's side had made a £43m ($55.5m) offer to bring the midfielder to Stamford Bridge. Citing sources close to the player, the report claims that no agreement has yet been reached.
According to French publication L'Equipe, the newly crowned Premier League champions have cooled their interest in Bakayoko and have reportedly switched their attention to Lyon midfielder Corentin Tolisso, with Kurt Zouma set to be used in a player-plus-cash deal.
Jose Mourinho is keen to strengthen his midfield in the summer and add more depth ahead of their return to the Champions League. The Portuguese manager currently has Michael Carrick, Ander Herrera, Paul Pogba and Marouane Fellaini as his options in midfield, but is looking for a defensive lynchpin, who will allow the likes of Pogba and Herrera to operate with more freedom in attack.
The MEN report also claims that Anthony Martial's presence at United could prove influential when Bakayoko makes a decision on his future owing to the close friendship between the two players, who played for a season - 2014/15 - together at Monaco.