The 26-year-old was a regular for Carlo Ancelotti's side during the 2021-22 campaign, making 42 appearances in all competitions, contributing 12 goals and two assists in the process.
Asensio's current deal is due to expire next June, though, and he has allegedly been told that he will not be offered a new contract by the reigning Spanish and European champions.
AC Milan continue to be credited with an interest in the Spain international, while Liverpool and Arsenal are both believed to be keen to bring him to the Premier League this summer.
According to Tuttomercatoweb, Real Madrid have informed interested parties that Asensio is available for between €25m (£21.6m) and €30m (£25.8m) in the current market due to his contract situation.
The attacker has scored 49 goals and registered 24 assists in 235 appearances for Real Madrid since arriving from Mallorca in 2014, winning three La Liga titles and three Champions League crowns.
Asensio has struggled for consistency since suffering a serious knee injury in a pre-season clash with Arsenal in July 2019, though, and Los Blancos allegedly now view him as surplus to requirements.
Rodrygo and Federico Valverde are both seemingly now ahead of him when it comes to the right side of the Real Madrid attack, which could make it difficult for the Spaniard to earn regular starts next term.
According to AS, Asensio's agent Jorge Mendes is pushing for an exit from Los Blancos, and Arsenal are viewed as the favourites to secure his signature.
Liverpool boss Jurgen Klopp is also believed to be an admirer of the Spaniard, though, with the Reds considering making a move before the end of the transfer window.
"Renew or leave Real Madrid? I think there are three possibilities, there is also the option to stay and fulfil my contract year," Asensio told Marca when recently questioned on his future.
"My future does not have to depend on whether or not I will have a place in the next World Cup. Rumours are part of football, and more so at Real Madrid. High-level players are always associated with it. Many players come out every summer and it's something we have to live with."
The attacker is not certain of a spot in the Spain squad for the 2022 World Cup, but he was involved in their UEFA Nations League fixtures in June, providing assists in their two matches with the Czech Republic.
Milan are also believed to be keen on the attacker, but the Italian outfit would struggle to match the wages that Arsenal and Liverpool could both offer him to move to England.