The 21-year-old is concerned by his lack of playing time at the Bernabeu under manager Zinedine Zidane.
And Spanish outlet Don Balon claim Asensio has stood up to Real president Florentino Perez and told him he will move to Chelsea if he isn’t given more opportunities to shine.
The forward had a magnificent start to the season, netting twice in the Spanish Super Cup win over Barcelona.
But since then he has found his chances limited and has had to make do with a place on the bench more often than not.
What makes the decision to not use Asensio even more surprising is how much Real have struggled in recent weeks.
The La Liga holders trail Barcelona by 10 points after 12 matches.
And with Karim Benzema and Cristiano Ronaldo both struggling for form Asensio would seem like the most likely man to offer something different.
Chelsea are looking to further strengthen their squad with the Spanish starlet a marquee name which would appeal to the Blues.
Antonio Conte’s summer signings have divided opinion with Chelsea struggling in their attempts to retain the Premier League.
Alvaro Morata, Tiemoue Bakayoko, Davide Zappacosta and Danny Drinkwater all joined before the transfer deadline.
The Blues currently sit third in the top flight, nine points behind leaders Manchester City.
Asensio has made 12 appearances in La Liga this campaign, scoring four times.