The Spain international, 23, has seen his performance levels plateau during 2018-19, in a difficult season for Real Madrid.
Los Blancos have gone through three managers during the course of the season, with ex-boss Zinedine Zidane returning after the failures of Julen Lopetegui and Santiago Solari.
Under all three, Asensio has struggled to hold down a regular starting spot; in 42 appearances he has started 23 times, completing the full 90 minutes on just 11 occasions.
And while he has a contract at the Santiago Bernabeu until 2023 - with a €700million release clause - Mundo Deportivo reports that Jurgen Klopp will make a move for Asensio this summer.