The reality is that Mbappe does not want to extend his contract at PSG, while PSG's own board management has offered him a new contract. I've heard that on one of the existing news channels, but I don't know if it's legitimate news or not because of course PSG will regret it if Mbappe is released on a free transfer later. Anyway, even though Mbappe has remained at PSG throughout this UCL season, I personally am still not sure that PSG will go further, because their competitors this season can also be said to be better.
The reality is Mbappe does not satisfy with his career in PSG. He see the club has opportunity to win Champions League because he has experience to know that the difference between champions and losers are small in some cases. With his experience in France national team, he should know it.
Moreover, he does want to win personal titles which he thinks he can not get such in PSG. It are probably one of biggest reasons and motivation why he does want to play in Real Madrid. The club has many super stars who win many Ballon d'Or and other personal titles. PSG is not a club at such level and has such tradition.
Mbappe only wants to serve with the existing contract with PSG while the offer of a new contract is a long decision if PSG can persuade Mbappe to stay then that is better, but if Mbappe is released on a free transfer then it is ridiculous that PSG has wasted so much money.
If in the UCL, PSG have to submit to Madrid then the possibility of this dream of being a champion will fade, even I have doubts about Madrid.
Actually Mbappe must be ready with his choice.
Beating Real Madrid or winning one Champions League can not change all history. Chance to win 3 Champions League like Real Madrid and Zizou is very small. It takes Chelsea many years from their first Champions League trophy to their second trophy. Chance to win a few Champions League in 5 or 10 years is low because it is scarce to see a club like Barcelona with Pep, Messi, Xavi, Iniesta, etc. years ago.
So I believe even if PSG win Champions League in the end of this season, Mbappe will still have more motivation to join Real Madrid rather than sign a new contract with PSG.