Los Blancos, having crashed out of the Champions League at the round of 16 stage - with a possibility of going trophy-less this campaign are hoping to strengthen their squad for next term.
However, SunSport reveals that the World Cup winner is planning to win the UCL title with the French League giants before considering a move away from the club.
Mbappe has scored 26 goals and six assists in 26 Ligue 1 appearances for the Parisians this campaign, and has been on the radar of the Galacticos since the exit of Cristiano Ronaldo last summer.
Madrid re-hired coach Zinedine Zidane this month and are planning to hand him a cheque of about £240 million to pursue the French star
Meanwhile, PSG might be forced to allow their man leave in a bid to comply with the UEFA Financial Fair Play regulations.
Reports from the camp of the 20-year-old confirmed that Mbappe's father and agent Wilfried have been in "constant talks" with Real Madrid over a summer move for his son.
But, the Parisians early ouster from the European elite competition after losing to Manchester United is believed to have swayed the forward's mind.
As he is hoping to lead the generation of stars who will bring the UCL title to France, after which he could join any other club interested in him.
"I still believe in this project, and I'm sure we'll do something good at this club ."
Mbappe has even been tipped to become the next Ballon d'Or winner amid his spectacular performances.