The Reds are believed to have shown an interest in the Spain international over the summer, but Los Blancos boss Carlo Ancelotti prevented the 25-year-old from leaving the club.
Asensio scored three times in Real Madrid's 6-1 win over Mallorca on September 22 but has otherwise found it difficult to make his mark for the capital giants this season.
According to ABC, the Spaniard is back 'on the market', and Liverpool are considering launching a move when the winter transfer window opens for business.
The report claims that Asensio is 'pondering' his future with Los Blancos, but Jurgen Klopp's side are likely to face competition from a number of clubs for his signature.
The former Mallorca youngster has scored 40 goals and registered 22 assists in 200 appearances for Real Madrid since arriving at the Bernabeu in July 2015. body check tags ::