The France international has spent the past four seasons at the Parc des Princes but is due to become a free agent at the end of the 2021-2022 campaign.
Mbappe is no closer to agreeing a new contract with the Ligue 1 giants, despite the recent high-profile addition of Lionel Messi to a star-studded attack that also includes Neymar.
PSG president Nasser Al-Khelaifi recently stressed PSG do not need to cash in on Mbappe, while Mauricio Pochettino last week suggested he expects the Frenchman to stay.
However, widespread reports from Spain and France indicate Madrid has now tested the Parisiens' resolve by tabling a big-money bid for the World Cup winner.
Reports in France claim PSG have rejected the offer, which would make it the third most expensive transfer ever, behind the €222m PSG paid Barcelona for Neymar and €180m handed over to sign Mbappe from Monaco in 2018 after a year on loan.
Mbappe has made 174 appearances for PSG in all competitions since arriving from Monaco in 2017, helping the club to 10 trophies, including three Ligue 1 crowns.
He has scored 133 goals since making his debut in September 2017, placing him behind Cristiano Ronaldo (144), Messi (162) and Robert Lewandowski (182) for goals in all competitions among players in Europe's top five leagues during that period.
The 22-year-old has assisted a further 58 goals for his team-mates, while only Lewandowski (39) has been directly involved in more goals in the Champions League than Mbappe's 37.