The 26-year-old's long-term future at the Santiago Bernabeu is believed to be uncertain as he has now entered the final 12 months of his contract.
Asensio recorded 12 goals and two assists in 42 appearances for Los Blancos last season, but he only started 23 times under Carlo Ancelotti and is now seeking regular first-team football as he approaches the prime years of his career.
According to TuttoMercatoWeb, via HITC, Asensio is 'desperate' to leave Real Madrid and is prepared to take a pay cut to help boost his chances of a summer exit.
Liverpool, Arsenal and AC Milan are among a number of top European clubs who have been credited with an interest, with the former looking to replace Sadio Mane who has joined Bayern Munich.
Even though Asensio is out of contract in 2023, it is thought that Real Madrid value the winger between €25m (£21.6m) and €30m (£25.8m).