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The 22-year-old is in the final year of his deal with PSG and has been linked with a move to Spain for years.
Marca have claimed Mbappe has told manager Mauricio Pochettino he has no desire to renew his contract.
It is also said that Madrid have promised the World Cup winner an eight-digit sum should he wait for a move.
Pochettino told Le Parisien: "Did Mbappe tell me he won't extend? These are private conversations. But in no case did Kylian talk to me about that. Can we imagine him leaving for free? I don't use imagination."
Madrid currently don't have the finances to sign Mbappe outright, who PSG are desperate to keep hold of.
Mbappe hasn't publicly said he wants to leave and even admitted his dream is to win the Champions League with PSG earlier this year.
In a joint interview with Neymar, the pair spoke to PSG's official magazine about their future goals and aspirations within the game.
During his exchange with the Brazilian, Mbappe said: "I'm not very good as a journalist. What would I ask Neymar? What is your big dream for you after everything you've achieved?"
Neymar responded: "My biggest dream with PSG is to win the Champions League and with Brazil to win the World Cup. And for you, Kylian, is it to win the Champions League, right?"
Mbappe then said: "My biggest dream is to win the Champions League with PSG, that would be fantastic."
However, Mbappe hasn't indicated to president Nasser Al-Khelaifi or sporting director Leonardo that he wants to stay.
The World Cup winner scored 27 times in 31 Ligue 1 appearances last season as PSG finished second, shockingly losing out on the title to Lille.
Meanwhile, Madrid signed David Alaba from Bayern Munich earlier this summer on a free transfer and are hoping to use a similar method.
Mauricio Pochettino: "Did Mbappé tell me he won't extend? These are private conversations. But in no case did Kylian talk to me about that. Can we imagine him leaving for free? I don't use imagination", he told to Le Parisien. :flag_fr: #PSG- Fabrizio Romano (@FabrizioRomano) July 29, 2021
Outgoings will help give Carlo Ancelotti funds and they've already let Raphael Varane join Manchester United, while Sergio Ramos has joined PSG.
Do you think Kylian Mbappe will join Real Madrid?
Let us know in the comments.