Jurgen Klopp has enjoyed a productive summer transfer window so far, adding as many as three players to his first-team squad at Anfield.
As well as Darwin Nunez joining in a deal which could rise to £86m, Fabio Carvalho and Calvin Ramsey have also arrived from Fulham and Aberdeen respectively.
There has been talk that Klopp may not make any further changes to a squad which challenged on four fronts last year, lifting two pieces of silverware and finishing as runners-up in the Premier League and Champions League.
However, according to Deportes COPE reporter Melchor Ruiz, the Merseyside giants are looking to complete a deal for Asensio over the coming weeks.
The 26-year-old has spent much of his senior days at the Bernabeu, contributing 49 goals from 235 appearances in all competitions.
Asensio has fallen down the pecking order under Carlo Ancelotti, however, with 12 of his 31 La Liga outings during the most recent campaign coming from the substitutes' bench.
With just one year remaining on the playmaker's contract, there is seemingly a reluctance in both camps to sign extended terms at Los Blancos.
While there is a willingness from Asensio to move to a new club, it is been widely claimed that he only wants to join a team where he will be first choice.
At this stage of the transfer window, that may not be the case at Liverpool with Mohamed Salah, Nunez, Diogo Jota, Luis Diaz and Roberto Firmino already competing for three positions.
The same report claims that Asensio has also received an offer from AC Milan, who were first linked with him ahead of the January window.
Earlier this week, it was suggested that Manchester United and Chelsea were pushing for the player's signature as they bid to strengthen their own squads.
Real Madrid are said to be looking for a fee in the region of £35m, while Asensio allegedly wants to earn in the region of €7m (£6.04m) per year at his next club.
However, both parties may have to be open to a compromise in order to facilitate a transfer before the end of August.