United are keen to sign a winger during this transfer window and, while their first choice is Jadon Sancho, Dembele represents a strong alternative.
The 23-year-old was a club-record signing for Barcelona when he arrived from Borussia Dortmund for €105m (£95.75m) in August 2017.
He has won two La Liga titles and the Copa del Rey during his three seasons in Spain, but has been restricted to 75 appearances in all competitions by a series of injuries.
On Tuesday, Manchester United had a bid for Jadon Sancho worth up to £91.3m (€100m) rejected by Borussia Dortmund.
The offer was for an initial £73m, rising to up to £91.3m, which fell short of Dortmund’s £108m valuation.
Dortmund set United a deadline of August 10 to agree a deal for Sancho, and have stated on several occasions that the England international will not be leaving now that date has passed.