Have you heard of the news about Liverpool's possible offer for Mbappe in the near future? It is said that Liverpool will offer 300 million euros for him. I think PSG would be really willing to sell him in case there is an offer like that. Because even though PSG have to pay a lot of money to Mbappe in case of selling him early they will still be in a serious profit.
It is much better than letting him go without a transfer fee surely. In this case I assume that Liverpool wouldn't be the only team to make the same offer. Real Madrid have been monitoring his situation for a long time. I would expect them to meet the demands of PSG and Mbappe both undoubtedly.
In this case I don't think Liverpool's chance would be higher to sign him.
I see no team better for Mbappe than Madrid. Madrid are team that don't rush to sign players but once they have set you as there target then it will be very hard for y not to be there player. I still blame PSG why you should keep player that has shown you clearly that he is not willing to stay anymore but rather you persuaded him to the extent of paying more than what he deserves.
The kind of contract pen down by Mbappe that PSG are paying now is very high and at the end he will still leave because his contract expires by 2024 and he's yet to sign another contract. Mbappe's dream is too big that if he remain in PSG he can't actualise it and the best place for young stars to actualise their dreams and be brought to more light is Madrid.
In Madrid getting UCL and Balloon d'Or will be much more easier for him. By 2024 he will be going to Madrid for free which is quite unfortunate, so PSG can't make a dime from his departure.