The French champions have been interested in the 26-year-old for a long time, and, according to sources, the forward has already agreed a five-year contract at PSG.
Dembélé has a €50 million ($55m) release clause in his Barca contract that is valid until July 31, when it goes up to €100m.
The France international also has an extension offer from Barca on the table, but PSG are confident that the player is more attracted by their project.
According to sources, there are still details to be sorted out but negotiations between the player's agent, Moussa Sissoko, and Luis Campos, the PSG sporting director, are positive.
Sources added that Barca boss Xavi wants to keep Dembélé as he considers him a key player, but understands the club knows that receiving a big transfer fee will help them with their financial issues.
The Catalans would be unable to do anything if Dembélé agrees to the move and PSG pay the release clause.
The forward has scored 40 goals in 185 appearances for Barcelona and has won three LaLiga titles in Spain.
Sources have told ESPN that Real Madrid remain Mbappé's preferred destination if he leaves this summer and to expect progress on a potential deal to be made next month.
Information from Rodrigo Fáez and Moisés Llorens was included in this report.