The Spain international has been linked with a move away from the Bernabeu in recent months.
Asensio, 22, is desperate to get more game time at Real under Zinedine Zidane.
The attacker has started just nine La Liga games this time out, netting four goals and chipping in with four assists.
And according to Spanish news outlet Diario Gol, Asensio will leave Real this summer.
They claim the winger already contacted Liverpool and Chelsea over a possible transfer.
Diario Gol say Liverpool want Asensio as a replacement for Philippe Coutinho, while Chelsea are in for the forward with Eden Hazard’s Stamford Bridge future up in the air.
However, Real president Florentino Perez is desperate to keep hold of Asensio this summer.
Perez has told the attacker to hold out until the end of the season before making a decision on his Bernabeu future.
Chelsea star Hazard is a long-term target for La Liga champions Real.
The Belgium international was tipped to quit Stamford Bridge for the Spanish giants last summer but the winger broke his ankle, ending any hope of a deal.
