I think he might stay in PSG because the money is just too good to be avoided. But a player like him will surely be interested to play for a team like Real Madrid. and I really hope that he will play for Real Madrid because he is not going to be achieving too many things with PSG in the future in my opinion.
He isn't a new player in PSG, and Real Madrid is a rich club as well. So, I believe money won't be his reason to stay or move. He must consider playing at a bigger club where he can have the opportunity to get more prestigious trophies. Especially to get the UCL trophy that he never got in PSG. And Real Madrid is a perfect club if he considers getting more trophies, including UCL trophy as well. Since this opportunity may not come twice, I assume Mbappe will agree with the offer from Real Madrid.
Why do you think Mbappe has a better chance of winning the Champions League with Real Madrid? I am not sure that Real Madrid will continue to be so lucky every year and the club will be able to win the Champions League again in the near future. The club expects a global restructuring, in which there will no longer be such players as Modric and Benzema, or they will no longer be able to make such a significant contribution to the success of the team. I would also say that Mbappe's opinion could still be influenced by who will be the head coach of Paris Saint-Germain next season, but apparently the player has already made his decision or the management has already informed him of this name.
I don't think winning the champion's league is the reason Mbappe is deciding to join Real Madrid, I strongly doubt that. I think in the next two years, PSG will be able to win the UEFA Champion's League and Mbappe is pretty very young so his motivation could be playing for a bigger club in Europe with rich history and heritage.