That is according to the London Evening Standard, who claim the Red Devils have enquired about the France Under-20 international ahead of the January window.
United's scouts have been keeping tabs on the striker this season, but also face competition from Everton and Tottenham for his services.
However, the Premier League trio are likely to miss out on Dembele after he committed his future at Fulham last month.
He said: “It is funny and flattering but on a serious note I am focusing on Fulham and promotion to the Premier League. All the other things do not affect me that much.
“I don’t deceive. I give 100 per cent every time I wear the shirt. For the rest, my agents are dealing with it.”
The 19-year-old, who has scored eight goals in the Championship this term, has also been offered a new four-and-a-half-year deal to stay on at Craven Cottage.