I know if Barcelona doesn't want to sell Gavi, he can push for a move if he doesn't want to play for Barcelona again, but no one knows if Gavi wants to leave Barcelona for another club now. If Gavi wants to leave Barcelona, nobody knows the club he has in mind to play next, and PSG being the only successful club in Ligue 1 is not enough reason to be sure that Gavi will choose PSG as his next destination because he will weigh other options to see the club that would be perfect for him.
It seems that there is no problem for Gavi to stay in Barcelona and I think the long-term project built by Barcelona will give more success for Gavi if he stays. But what he needs to do is how to get a place as a core player because there may be a lot of competition for the midfield in Barcelona so it takes the best ability for Gavi to enter as a core player. PSG is indeed one of the successful teams in Ligue 1 and maybe the trophy they got in the Champions League this season will give added value to the player who decides to play there.
No one knows what Gavi's future will be like because Barcelona themselves probably won't sell him. I think Barcelona themselves are much more interested in keeping Gavi because he is still quite young and probably has good long-term prospects for the club's journey.