Both Spanish publication 'Sport' and English counterpart 'The Daily Mirror' report that the 'Blues' are plotting a summer move for the starlet.
Their struggles this season have shown the Chelsea board that the club needs fresh blood and they see Asensio as the perfect man to freshen up their squad.
As a result, Antonio Conte's side are reportedly ready to offer up to £120m (€136m) for the former Real Mallorca man.
With Asensio currently tied down until 2023 and with his release clause set at €500m, 'Los Blancos' find themselves in a strong bargaining position, hence the high mooted fee.
Chelsea do have an ace up their sleeve though in the form of Eden Hazard. Everyone is aware of Real's reported interest in the Belgian and it is thought that Chelsea could finally let their star man leave, providing Asensio were to come the other way as part of the deal.
The main problem for the English side would be convincing the Spaniard to swap Madrid for London and showing him that his chances of regular first-team football would be boosted by a move to Stamford Bridge.
With the January transfer window having only just closed, there remains a long time until the summer window will open, but as things stand it appears that Real would not stand in the 22-year-old's way should he decide to leave the club.
Asensio has yet to comment on either the rumours or his lack of game time, but that could change in the coming weeks and months.