CNN has not been able to independently verify L'Equipe's or Le Parisien's reporting.
PSG sporting director Leonardo said the club have turned down an initial bid from Real Madrid.
"Yes, we said no verbally to Real," Leonardo told RMC Sport. "No one is being held back. If someone wants to leave and our conditions are met, we'll see."
"The offer is considered to be a far cry from what Kylian is worth today. We also owe part of this money to Monaco and we consider that the bid is not sufficient.
"Our position has always been to keep Kylian, to extend his deal, that has always been our goal and still is."
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.
Earlier this month, PSG president Nasser Al-Khelaifi said he hoped the arrival of Lionel Messi at PSG would convince Mbappé to stay at the club.
"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."