Spanish outlet Don Balon, as quoted by the Daily Star is reporting that the west London side are desperate to tie up a deal to sign the 22-year-old Real Madrid starlet this summer.
The same report says that Asensio is anxious to get more playing time after having struggled to hold down a spot in the first team at Real Madrid under Zinedine Zidane this season.
The attacker has scored six goals and made four assists in La Liga this season but has been limited to just 14 starts in the top flight for the Spanish side.
According to the same story, Chelsea FC striker Alvaro Morata has been working to convince the west London side to go all out and try to sign the Spanish attacker this summer.
However, the same report says that Real Madrid have no intention of selling the attacking midfielder this summer, suggesting that Chelsea FC are going to struggle to be able to conclude a deal for Asensio.
Meanwhile, Chelsea FC are preparing to return to Premier League action on Sunday afternoon when they welcome top-four rivals Tottenham Hotspur to Stamford Bridge.
MORE: Man United latest news
MORE: Arsenal latest news
MORE: Chelsea latest news
MORE: Liverpool latest news