Asensio, who is currently with the Spain squad at the 2018 World Cup, scored 11 times in 53 appearances in all competitions for Los Blancos during the 2017-18 campaign.
The 22-year-old has a contract with his Spanish club until the end of the 2022-23 campaign, and it is understood that his release clause stands in the region of £615m.
However, according to Mundo Deportivo, Liverpool are prepared to smash the British transfer record to sign the attacker, with an offer of £150m already submitted to the European champions.
The Spain international joined Madrid from Mallorca in 2014, and has scored 21 times and registered 10 assists in his 91 appearances for the capital giants in all competitions.
Asensio's future at the Bernabeu is believed to depend on whether Los Blancos make a move for Paris Saint-Germain forward Neymar, who continues to be linked with a return to Spain.