Mbappe is eager to seal a dream move to the Spanish giants this summer after turning down more than three different proposals from PSG to extend his stay in recent months.
With the 22-year-old having less than a year left on his contract with the French giants, PSG will could lose him for nothing should they keep him at the club.
It was assumed that the arrival of Messi could have been a good business to keep Mbappe at the Parc de Princes but the French forward is still pushing for a move away from France.
Mbappe joined PSG from Monaco on an initial season-long loan in 2017 before joining permanently for €180 million the following year.
Reports are that PSG are considering Real Madrid's offer presented on Sunday on whether to sell or not.
Since arriving at PSG, Mbappe has scored 133 goals in 174 games for the club in all competitions.
Despite Madrid battling with €300 million in losses due to the coronavirus pandemic, the White Angels are willing to cough-out €180 million for their number one target for the summer.
PSG new recruit Lionel Messi is expected to make his debut against Reims and is hoping to play with Neyamr and Mbappe.
“PSG have made spectacular signings on top of the players that were already here,” said Messi.
“I’m eager to get playing and I’m going to be doing it alongside some of the best players in the world. It’s going to be a marvellous experience every day. I hope I can continue to win titles. I have the same hunger as when I was younger.”