CNN has not been able to independently verify L’Equipe’s or Le Parisien’s reporting.
The 22-year-old World Cup winner joined the Parisian club from AS Monaco initially on loan in 2017, before the deal was made permanent in 2018.
The Frenchman’s current contract with PSG expires on June 30, 2022 and he is yet to sign a contract extension.
Neither PSG, Real Madrid nor Mbappé’s lawyers were immediately available for comment when contacted by CNN.
Mbappé has scored 133 goals in 174 games for PSG and was the top goalscorer in Ligue 1 last season with 27 goals.
“I think everybody knows the future of Kylian — he wants to win, he wants to win trophies,” Al-Khelaifi told a press conference.
“He said it in public, he wants to have a competitive team and we have the most competitive team in the world. There is no excuse for him now — he can’t do anything else but stay.”