I think it doesn't make much sense for Real Madrid to wait for next season and Mbappe becoming a free agent. In any case, they will be forced to pay him a signing bonus because many clubs will offer this to Mbappe. If PSG now reduces the amount of compensation to a reasonable level (I think it's 120-140 million, no more), then Real will buy Mbappe now and not in a year.
The current situation with Mbappe's transfer rumors is similar to that of Luis Figo joining Real Madrid in 2000 and we can see the same words spoken by club president Florentino Perez at that time. Maybe no one would have thought that Luis Figo, who was previously at the top of the top when defending Barcelona, but with certainty Florentino Perez can bring him in to fulfill his campaign promises after becoming club president. I think the same thing will happen with Mbappe's transfer this time and indeed Real Madrid seems to be deliberately bidding for Mbappe at the end of August. With that, PSG can't do much and they have to be willing to lower Mbappe's fee so that it's cheaper to agree.