I don't think Real Madrid are the only team most likely to sign Mbappe if PSG want to sell him. Mbappe could move to a Premier League team if they offer value that is hard enough for PSG to refuse. You have to know that Real Madrid didn't want to pay Mbappe big money to PSG because apparently they just wanted to sign him for free.
The question is which Premier League team has got the balls to currently offer PSG about 200M 
Manchester City is definitely out of the question for now.
By the way, at one point Real Madrid offered PSG something like 200M or even over the value, but PSG was too stubborn to sell him off due to ego issues. I don't see why you said Real Madrid wanted to sign him for free. That was just their last option after PSG refusal of all the offers.