Mbappe should absolutely be sold this summer if PSG don't want to lose him for free. They seemed very determined about not letting him go like that. On the other hand Mbappe is very determined to leave the team at the end of his contract this time. It doesn't look like we will have a similar scenario like Mbappe will be offered another enormous deal and he will take it. He really looks like he wants a new adventure from now on. That adventure is most likely to be with Real Madrid.
Real Madrid don't want to push this deal for now while they have a chance to sign him for free in six months. Let's see what next days will bring to us.
I think they don't want to sell Mbappe but PSG wants to somehow earn more money from Mbappe than this by extending his contract and selling him for more money, or even if they wanted to keep Mappe in the club for a longer period of time and benefit from the player who can help Enrique achieve his goals in the club.
It's clear PSG don't want to sell Mbappe, it's the attitude shown by Mbappe that forces PSG to sell him immediately instead of letting him go as a free transfer next season. PSG just lost Messi who chose to join Inter Miami, Mbappe's departure will further complicate PSG situation because they lost two important players at once.
PSG will certainly find it difficult to find an ideal replacement for them, especially since the stock of quality strikers available on the market is currently very limited. Enrique, who has just been appointed as the new coach, will experience problems in strategizing if the agreement with Mbappe reaches a dead end, he must make several changes to keep PSG's performance on track, without a goal machine that fills the striker position in the squad he leads, it is difficult for Enrique led PSG to success in the Champions League.