Barca president Josep Maria Bartomeu is understood to have concluded talks on a five-year contract for the forward on the Mediterranean coast on Thursday.
Barcelona are unlikely to confirm the deal until it is signed but Sky Sports News have been told the move is agreed in principle and could be worth as much as £110m.
Dembele has been a primary target for the Catalan side this summer, along with Liverpool midfielder Philippe Coutinho, as Barcelona look to bring in reinforcements following Neymar's £198m move to Paris Saint-Germain.
Borussia Dortmund sporting director Michael Zorc said last week Dortmund would not wait until the last few days of the transfer window for a deal to be agreed, insisting the 20-year-old would be staying at the club if their asking price was not met.
Dembele joined Dortmund from French Ligue 1 club Rennes last summer. He scored six goals and contributed 13 assists in 32 Bundesliga matches during his first season at the Westfalenstadion.