Transfers to be made in January transfer window could be effective enough to change the fate of the Champions League winner maybe. I can't imagine what would happen in case of seeing Real Madrid's signing a deal with Mbappe. However even if this happens he wouldn't join the team right away of course. They would need to wait for him until the end of the season unless they buy his clause to bring him immediately.
However truthfully I don't expect Real Madrid to sign him in January because probably he would like to wait for the end of the season to make his decision on his future.

I expect PSG to be eliminated early again which would make things really difficult for convincing Mbappe to stay.