Asensio also has less than 12 months left to run on his contract with the La Liga and Champions League holders, making him a likely candidate for the axe this summer.
Los Blancos supposedly have no intention of extending Asensio's stay at the Santiago Bernabeu, and the 26-year-old will therefore be free to leave for the right price in the coming weeks.
Liverpool, Manchester United and Arsenal have all been credited with an interest in Asensio, and according to Defensa Central, the Gunners have ramped up their efforts to sign the attacker.
A fee between €25m (£21.6m) and €30m (£25.8m) could supposedly be enough to prise Asensio away from the Spanish capital this summer, as Real Madrid are desperate not to lose the 26-year-old on a free transfer.
During his seven-year stay at the Bernabeu, Asensio has made 235 appearances for the capital giants in all competitions, scoring 49 goals and setting up 24 more in the process.
Asensio hit 12 goals and two assists in 42 games under Ancelotti last season, but he only started 19 La Liga games, and his opportunities in the Champions League were few and far between.
The Spaniard did not start any European fixture from the quarter-finals onwards, and he was an unused substitute in the 1-0 win over Liverpool in the final.
Should the former Mallorca youngster be allowed to head elsewhere this summer, Real Madrid are supposedly keen to replace him with Bayern Munich man Serge Gnabry.
Real Madrid would supposedly look to use the funds raised from Asensio's sale to strike a deal for the former Arsenal youngster, who is also due to become a free agent next summer.
Bayern are said to be after a fee of €50m (£42.5m) to part ways with Gnabry, and Real are keen to offload Asensio before making their move for the Germany international.