United assistant manager Steve Round, as well as coaches Nicky Butt and Phil Neville, were present to watch the 2-2 draw a fortnight ago to monitor Darren Fletcher's comeback from a serious bowel condition.
The Scotland international midfielder got through the match unscathed, and The Mirror reports that it was Dembele's performance that left a lasting impression on the Old Trafford staff.
The 17-year-old, who joined Fulham from Paris Saint-Germain in 2012, scored once, and his overall display led them to make comparisons between him and former Chelsea striker Didier Drogba.
A French youth international, Dembele has recently signed a contract extension at Craven Cottage until 2016, but that is unlikely to deter United from making a move for him in the future.
Dembele shares the same name as, but is not related to, former Cottagers midfielder Moussa Dembele, who was sold to Tottenham Hotspur in a £15m deal last summer.