Despite only turning 16 last August, the Frenchman has made 12 first-team appearances for Les Gones during the 2019-20 campaign, contributing three goals and two assists in the process.
Both Madrid and Barcelona are believed to be keeping a close eye on the teenager, who came through the youth system with Lyon and has a contract with the Ligue 1 giants until the end of the 2021-22 campaign.
However, according to AS, Los Blancos have first refusal on the youngster, placing them in a strong position to secure his signature should the French side agree to a sale.
Cherki has shown his versatility by operating as a right winger, left winger and as an attacking midfielder for Lyon during the current season.
Manchester United and Liverpool have also previously been credited with an interest.