United are not giving up on the 29-year-old, however, and have no interest in selling him in January.
Fellaini has become a regular under Jose Mourinho, scoring four goals from nine games in all competitions this term, and the manager is likely to put pressure on the club to ensure he stays.
That has not stopped the former Everton man from beginning his search for a new club and sees Turin as a possible destination.
Sources with the Italian side have told Goal that Fellaini's representative has discussed a potential deal, but has been knocked back as the Turin side are targeting other options.
Massimiliano Allegri's side are looking to add to their midfield in the near future, but they have no plans of going for Fellaini and are are instead prioritising Emre Can and 22-year-old Leon Goretzka - both of whom are also out of contract at the end of the season.
Although Juve director Giuseppe Marotta recently distanced the club from making a January move for Liverpool's Can, the Bianconeri will try to lure him away on a free signing in the summer.