That’s the incredible claim from Spanish outlet Mundo Deportivo.
Their report suggests that the Reds have followed up interest in Asensio with an official approach that would break the current British transfer record - currently the £142million fee that Barcelona paid to sign Philippe Coutinho from Anfield back in January.
Madrid play-maker Asensio is viewed as one of the stand-out prospects of the next generation for club and country - as reflected by the apparent £613m release clause put into the six-year contract he signed at the Bernabeu back in September.
The 22-year-old supplied 11 goals and six assists in 53 appearances in the season just finished and his form saw him named in Spain’s 23-man World Cup squad for Russia.
Asensio came off the bench as Spain overcame Iran 1-0 in their second group game on Wednesday.
But many of his appearances for Los Blancos have also been cameos from the bench given the quality of the European champions' front three and midfield, which Real demonstrated in the Champions League final against Liverpool in May - which they won 3-1. Asensio was an 89th-minute replacement for Karim Benzema in Kiev.
It is claimed that Asensio is frustrated with his lack of starts in the Spanish capital and while it is not yet known what role he will play under new Madrid boss Julen Lopetegui, formerly in charge of the national team, Liverpool want to pounce on his reported discontent.
That’s after a deal for Lyon captain and France ace Nabil Fekir fell apart at the last minute on the eve of the World Cup.
Fekir looked set to follow Fabinho and Naby Keita through Melwood's doors after completing a medical for a £53m move, but - over apparent hesitations on Liverpool’s side surrounding a past knee injury - the transfer collapsed.
Real Madrid, Mundo Deportivo claim, intend to reject the Merseyside outfit’s Asensio bid but will first study it rather than flat-out dismissing it.They add that Lopetegui’s opinion of Asensio’s ability means it will be difficult for Liverpool to seal any move for the talented youngster this summer, but that they are keen to try and force the issue.
Asensio has not been mentioned as a target by sources closer to Anfield, however, casting understandable doubt over the legitimacy of Mundo Deportivo’s report.
It is thought that a deal for Fekir is not yet completely dead, meanwhile, although the complex situation between Liverpool and Lyon means it is unclear whether another agreement can be reached.
Jurgen Klopp is seeking a No 10 to replace the creativity lost by Coutinho’s departure six months ago but no Fekir alternative has been reliably mentioned as Liverpool wait until the conclusion of the World Cup to press on with further business.