According to Spanish publication Don Balon, Chelsea owner Roman Abramovich is spearheading the move because he is determined to bring him to Stamford Bridge.
Abramovich has reportedly already begun work on a potential deal by sending representatives to discuss a switch with the 21-year-old.
Asensio is under contract at the Bernabeu until the summer of 2023 after putting pen to paper on an extension last year.


Despite emerging as one of ’s hottest prospects, Asensio’s progress has stalled in recent months.
He has chipped in with 10 goals and four assists in 44 appearances this campaign but has hit the back of the net just three times since mid-November
According to Don Balon, Alvaro Morata has talked up Asensio’s potential to Chelsea chiefs.
Morata played alongside Asensio at Real Madrid before heading to  last summer.
The Spain international previously attempted to quash talk of a move away from the Bernbaeu.


Speaking in February, he said: "I always try with everything I have left.

To reach the biggest club in the world where there are millions of fans and a lot of pressure was a brutal change.
“My debut was in a final. I was not nervous, I scored a goal and I won my first title.
In all my debuts I've always been able to score goals and I have come to Madrid to stay and score many goals."


According to Spanish publication Don Balon, Chelsea owner Roman Abramovich is spearheading the move because he is determined to bring him to Stamford Bridge.
Abramovich has reportedly already begun work on a potential deal by sending representatives to discuss a switch with the 21-year-old.
Asensio is under contract at the Bernabeu until the summer of 2023 after putting pen to paper on an extension last year.


Despite emerging as one of ’s hottest prospects, Asensio’s progress has stalled in recent months.
He has chipped in with 10 goals and four assists in 44 appearances this campaign but has hit the back of the net just three times since mid-November
According to Don Balon, Alvaro Morata has talked up Asensio’s potential to Chelsea chiefs.
Morata played alongside Asensio at Real Madrid before heading to  last summer.
The Spain international previously attempted to quash talk of a move away from the Bernbaeu.


Speaking in February, he said: "I always try with everything I have left.

To reach the biggest club in the world where there are millions of fans and a lot of pressure was a brutal change.
“My debut was in a final. I was not nervous, I scored a goal and I won my first title.
In all my debuts I've always been able to score goals and I have come to Madrid to stay and score many goals."