The France international's current deal in Paris is due to expire next summer, meaning that, as it stands, he will be free to hold talks with foreign clubs in January over a pre-contract.
Real Madrid failed with a late move to sign Mbappe during the summer transfer window but remain the favourites for his signature.
According to AS, the World Cup winner is edging closer to a move to the Spanish capital, with PSG close to admitting defeat in their bid to keep hold of him.
The report claims that the 22-year-old is desperate to secure a switch to Real Madrid and could agree a deal in January ahead of a move at the end of the 2021-22 campaign.
Mbappe has again been in impressive form at club level during the current season, scoring nine times and registering 12 assists in 18 appearances in all competitions.
In all, the Frenchman has netted 141 goals and registered 73 assists in 189 outings for PSG since making the move from Monaco. body check tags ::