Zinedine Zidane is set to embark on a major overhaul of Los Blancos' squad following the club's failure to win a major trophy this season.
The Frenchman has made compatriot Mbappe one of his leading targets in the summer market, and Madrid are willing to spend up to £240m on the World Cup winner.
That would comfortably surpass the £198m PSG paid Barcelona to secure the services of Neymar in 2017.
The Ligue 1 champions would be loath to lose the 20-year-old, who is widely viewed as a future Ballon d'Or winner.
However, PSG continue to be dogged by Financial Fair Play issues and selling Mbappe would alleviate some of the pressure forced on them by UEFA.
The former Monaco striker has scored 36 goals in 41 appearances in all competitions this term.
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