The 26-year-old was a regular for Los Blancos during the 2021-22 campaign, making 42 appearances in all competitions, contributing 12 goals and two assists in the process.
Asensio found it difficult to secure starts in the latter stages of the season, though, and his contract is due to expire in June 2023, which has led to a host of speculation.
A recent report claimed that Real Madrid have placed a £35m valuation on the Spaniard, with a number of clubs said to be interested in signing him ahead of the 2022-23 season.
Here, Sports Mole picks out three teams that could potentially sign Asensio during this summer's transfer window.
AC Milan are thought to be at the head of the queue for Asensio, with the Italian champions identifying him as a leading target during the summer market.
According to AS, Milan's strong relationship with Real Madrid has placed them in a strong position to bring the versatile attacker to San Siro ahead of the 2022-23 campaign.
However, a lot would still have to happen for the Italian side to sign him, including Los Blancos dropping their asking price to around £25.8m.
Milan also cannot compete with Premier League clubs when it comes to salary, meaning that the attacker would have to accept a lower wage packet in order to join the Italian champions.
The Serie A giants allegedly 'love' Asensio, though, and are prepared to offer him a key role in their starting XI in order to tempt him into joining the club.
Arsenal are looking to improve their attack during this summer's transfer window, and the Gunners are believed to have a firm interest in Asensio, who could be a cheaper option to Leeds United's Raphinha.
Mikel Arteta's side would not be able to offer the attacker Champions League football for next season following their fifth-place finish in last term's Premier League.
However, the chance to move to London could appeal to Asensio, and he could secure a starting role for the Gunners if he hits the ground running in the early stages of next season.
Arsenal would be able to meet the attacker's wage demands and should also be able to afford his transfer fee, but they will be competing against clubs that have Champions League football next term and are also genuine challengers for the league title in their respective countries.
Liverpool boss Jurgen Klopp is believed to be a firm admirer of Asensio, with the Reds already making contact with his agent to register an interest this summer.
Klopp is believed to be open to signing another forward this summer, and the chance to bring Asensio to the Premier League is said to appeal to the German.
However, it is unlikely that the Spaniard would be a key player at Anfield, with the Merseyside giants instead viewing him as a squad player capable of filling in across the front three.
Asensio has not been the same player since suffering a serious knee injury in July 2019, but he still has time on his side when it comes to potentially re-establishing himself as one of the most exciting attackers in European football.