If you are saying that Mbappe will join Real Madrid next year, that means he is going to join them as a free agent and psg would not get any money from his move there. Psg would not let that happen, that is why they are trying to force him to extend his contract, because if he does not extent it, there is a possibility that Mbappe who is the most valuable player in the world would leave for free, that will be a huge loss for psg, so i don't feel they will let it to happen. If Mbappe does not agree to the extension of his contract, psg may be forced to sell him this summer.
I feel Mbappe still wants to stay with PSG for maybe one or two more seasons and at the same time he wants to join Real Madrid. He might be staying back at PSG for now out of pity and all the good benefits he has gotten from the club. He knows that Real Madrid is the best place for him to win the Champions League trophy he has always wanted but wants to make his move there in a good term with his current club. He only has two choice now, either he extends his contract or he’ll be sold this transfer window in order for the club to avoid potential loss of the most valuable player now. It’s better he decides now and not have any regrets later.
Another rumor is related to the problem between PSG and Madrid reportedly the two teams are not in good condition so PSG prefers to release mbappe to another team anyway except Madrid while Mbappe himself prefers move to Madrid, looks like a complicated agreement here.
There disagreement with other should not stop this deal from working if at all this is the issue on ground about Mbappe’s move. PSG wants the money before they’ll release the player and Real Madrid has the money to offer them, so their disagreement shouldn’t come in line of business if the receiving club is ready to pay the amount slated on the player.