Dembele, 24, was a target for PSG in January, and his future has been the subject of constant speculation in recent months.
A deal for Dembele stalled three months ago with the Parisians unable to complete a move due to financial fair play rules.
According to Sky Sports, the player already had a verbal agreement to head to the French capital, with Chelsea, Manchester United and also Liverpool being offered the chance to sign him.
Barca officials had told him to find another club after refusing to accept new terms that included a pay cut.
Dembele, however, was a key figure in Barca's emphatic 4-0 victory over arch-rivals Real Madrid last weekend, and even sparked rumour he is prepared to stay beyond his current deal.
A video posted online showed Barca players celebrating the important win, whilst a distant voice was heard by fans which was transcribed as "Chicos me quedo!", translating to "Boys, I'm staying!"
For someone who was told to leave in January, Dembele has returned six assists and a goal in his last six outings representing a big upturn in form.
French outlet RMC Sport, though, have revealed chiefs at PSG will give the green-light for Angel Di Maria's seven-year spell at the club to come to an end this summer, enabling them to capture Dembele.
Di Maria, 34, is eyeing an emotional return to Benfica and, should he leave, it would free up space for them to move for Barca's out of contract winger.
The Catalans are prepared to allow their star to remain at the Camp Nou with vice-president Rafa Yuste saying: "If Ousmane Dembele continues to feel comfortable, we can meet his agents and he could stay."
But Dembele is likely to be persuaded by PSG's offer and the opportunity to link up with the likes of Lionel Messi and Neymar in the French capital.