The former Espanyol star, who signed a new six-year deal with the Madrid side in September, made 53 appearances for Real last season, scoring 11 goals and assisting in a further six strikes.
According to Telemadrid, Liverpool are now willing to use all of the money they received for Coutinho and add a little on top to make a bid of 180 million euros for Asensio, which would convert the 22-year-old into the second-most expensive player of all time ahead of Coutinho.
The Metro, via Spanish publication Sport, claim there will be a mass clear out after the World Cup with Real's new manager Julen Lopetegui considering Asensio as one for the transfer list.
The Reds hold a long-standing interest in Asensio, having also attempted to sign him last summer.
Liverpool's interest in Lyon attacker Nabil Fekir was well reported, with the club failing to finalise a deal for the Frenchman.
Inching closer to Liverpool?
In the main, he's been used as a substitute for Real Madrid, with three of Cristiano Ronaldo, Isco, Gareth Bale and Karim Benzema lining up as an attacking trident.
The links to Asensio will understandably excite Liverpool supporters, as the prospect of him alongside the likes of Mohamed Salah, Roberto Firmino and Sadio Mane would be frightening for the rest of the Premier League.
Therefore, Liverpool seem to be on stable ground as they look to land Asensio this summer.
Links with the likes of Neymar and Eden Hazard refuse to disappear, meaning Real Madrid may need to raise a large amount of funds.