The 26-year-old faces an uncertain future at the Santiago Bernabeu as he will soon be entering the final year of his contract.
However, the report adds that Asensio would favour a move to Serie A champions AC Milan, where he would be guaranteed more regular first-team football.
The report also claims that Milan are pushing hard to sign the Spain international, who is earning around €6m per season with Los Blancos, but they may struggle to compete with Liverpool financially.
Asensio, who has also been linked with Arsenal and Manchester United, recorded 12 goals and two assists in 42 appearances for Real Madrid last season, and he is allegedly valued by the Spanish giants at £35m. body check tags ::