He was limited to 28 starts in all competitions last term and made just a cameo appearance in the Champions League final against Liverpool.
According to Diario Gol, who are renowned for their outlandish transfer rumours, Chelsea are eager to bring Asensio to Stamford Bridge.
It is claimed that Stamford Bridge bosses are pushing hard to secure his signature.
They have reportedly tabled a contract offer that Asensio would find difficult to reject.
The 22-year-old is reportedly eager to find out if he will have a regular role next season.
He is likely to be promoted in importance if Cristiano Ronaldo, Karim Benzema or Gareth Bale are sold.
He has previously claimed he would snub a move away from the club if the opportunity arose.
Speaking last December, he said: “I am at the best club in the world and I am happy at Real Madrid.”