Florentino Perez said that Real Madrid were going to sign Mbappe but not this summer. However the coming days might still change this situation.
Big change firstly comes from Real Madrid. They did not plan to lose Benzema in the summer and were in talk with Benzema to sign a new contract. However, the super good offer from Al Itihad convinced Benzema to leave and Real Madrid respected the player to end the contract one year before its expiration with a consensus between Real Madrid and Benzema.
It changed many things in Real Madrid transfer strategy and they will have to look at Mbappe one year sooner than they thought.
Because as you said PSG board are determined about not letting him go for free. But while there is no chance to convince Mbappe after that statement of him there is only one choice left.
PSG will need to sell Mbappe as soon as possible. Otherwise they would have no chance left after the new season starts. Because he will be much closer to be available for contract offer from other teams. Honestly PSG owner is very rich so they shouldn't worry about letting him go for free.
PSG know that they should not put themselves at a similar situation like one year ago, when Mbappe became a free agent and can leave Paris without any cent for the club. If they can not prolong the contract with Mbappe in this summer, they will let him go.
But again I read rumor that the French President Macron tries to do some political pressure to keep Mbappe in PSG and Ligue 1. Will it happen again and Mbappe will stay in PSG with a new contract?
I know that they are paying a huge amount of money for Mbappe's contract now so they want a consolation amount against that. But they really need Mbappe for being competitive for the Champions League next season. I don't know if they can find a great replacement in a short time.
Many comparison between Mbappe and Haaland as new dual competitors like Messi and Ronaldo. This season, Haaland got a treble achievement with Manchester City and won his first Champions League. It will be biggest challenge for Mbappe to decide his future in PSG. He is not happy because Champions League is his dream some recent seasons but never touched it.