The 21-year-old has been at the top of Madrid's transfer wishlist since joining PSG from Monaco, but his exorbitant price and his club's unwillingness to sell one of their star assets has so far made a deal impossible.
However, Mbappe's current contract at the Parc des Princes expires in 2022, and Marca reports that they would have to sell him next summer if they cannot agree on an extension.
PSG remain desperate to keep the World Cup winner at the club and are expected to offer him a lucrative new deal in order to extend his stay, but Madrid are optimistic that he will reject those advances in order to force through a switch to the Spanish capital.
Any deal in the summer of 2021 would likely see his fee drop with only one year remaining on his contract too, although Mbappe is still expected to command one of the most expensive transfers in history.
The report claims that PSG know they have one year left to convince Mbappe to stay at the club, and otherwise will cash in on him next year rather than risk losing him for free in 2022.
Mbappe scored 30 goals and set up 17 more in just 33 appearances for PSG in 2019-20 before the domestic season was curtailed.