The French forward could be on the move in 2021 with his contract due to expire a year later.
PSG are braced for a bidding war if they fail to persuade Mbappe to extend his contract before then as several clubs queue up for his signature.
The 21-year-old has praised Liverpool’s success previously, while Man City and Man Utd are among the teams keen on landing his signature.
Reports have suggested City could even be prepared to sign Mbappe rather than making a move for Lionel Messi.
However, according to journalist Josep Pedrerol, Mbappe has already told PSG he wants to move to Madrid.
Pedrelol explained on El Chiringuito TV how Mbappe dreams of playing in the white of Madrid and believes a move to the Bernabeu could help accelerate his ambitions of winning the Ballon d’Or.
Key to Mbappe’s decision is also the presence of a number of French players in Madrid’s ranks, as well as manager Zinedine Zidane, who the striker holds in high regard.
PSG remain hopeful they can convince Mbappe that his future lies at the Parc des Princes after a season in which the club reached the Champions League final.
But it seems he has already made his mind up to further his career elsewhere.
In a related story, a football club in Germany has suffered a 37 -0 demolition.