Chelsea are planning to make things more heated for signing Mbappe. They are thinking of joining the race seriously soon. However their offer is rumoured to be a cash plus player kind of one. But PSG are really desperate to sell him as long as they get a reasonable offer. They would accept Chelsea's offer if they don't keep the cash at a low level.
Mbappe would refuse to speak/negotiate with Chelsea, like he did to the Saudi Arabian club Al Hilal that wanted to sign him, even though PSG had already agreed the fee that Al Hilal brought to the table. I mean, Chelsea are not even in the CL, and it does not seem that Mbappe would want to play for Chelsea, even if it is a loan deal. Mbappe is making things very hard for PSG, even if they agree on a fee with another club, there is no guarantee that Mbappe will agree to negotiate with the club.
I think Mbappe is ready to stay in PSG for this season notwithstanding whatever is to come and the treatment he will get, he has decided to move to Real Madrid, thus it is either they come and buy him this summer or he stays in PSG and joins them next summer for free.