The Frenchman's current deal only expires in 2022 but there have been no indications he will sign an extension in the French capital.
Mbappe has been linked with the likes of Real Madrid and FC Barcelona over the past couple of years as his stock continues to rise with some impressive performances.
However, it seems the 22-year-old has made up his mind about his next club as Spanish outlets Cuatro and Cadena SER report that he wants to join Los Blancos.
Los Blancos were reportedly among the teams who wanted to sign Mbappe when he was still at AS Monaco, but they have maintained their interested in the last few years.
Madrid Zinedine Zidane was asked about Mbappe ahead of Saturday's El Clasico, but did not want to add to much on his countryman.
"I know him. We know each other well, but he's not my player so I can't say anything. He is a great player and we'll see what he wants to do in the future, but that's not my concern right now," Zidane said, according to Goal.
