Asensio, 22, is desperate to get more game time under Zinedine Zidane at Real.
And according to Spanish news outlet Don Balon, who often make sensational transfer claims, Chelsea have tabled a proposal for the attacker.
They say the Blues have offered £132m (€150m) for Asensio ahead of the summer transfer window.
Chelsea ace Alvaro Morata, who quit Real for Stamford Bridge last summer in a £70m deal, has convinced Blues transfer chiefs to go all out for Asensio at the end of the season.
Perez feels Asensio is Real’s future, especially with Gareth Bale and Cristiano Ronaldo’s transfer plans up in the air.
Asensio has been in fine form for Real this time out under Zidane.
Chelsea are back in action on Sunday afternoon when Antonio Conte’s men host Tottenham in the Premier League (4pm kick-off).