The Spain international continues to be heavily linked with a move away from Bernabeu, with his current deal due to expire next summer.
Arsenal, Liverpool and AC Milan are among the clubs to be linked with the attacker, but Newcastle have allegedly recently joined the race for his signature.
According to journalist Ekrem Konur, Eddie Howe's side have launched a £20m bid for the 26-year-old, but it is believed to be at least £5m short of Real Madrid's asking price.
Asensio was a regular for Los Blancos during the 2021-22 campaign, scoring 12 goals and registering two assists in 42 appearances in all competitions.
Meanwhile, according to journalist Gianluigi Longari, Everton are ready to explore the possibility of signing Tottenham Hotspur defender Japhet Tanganga this summer.
Longer claims that Frank Lampard's side are keen to take the versatile defender on loan for the 2022-23 campaign, with Clement Lenglet's recent arrival at Spurs likely to push him down the pecking order.
Tanganga came through the youth system at Tottenham, and he has made 43 appearances for the North London club in all competitions, including 19 outings during the 2021-22 campaign.
The 23-year-old was a regular in the first half of last season but was absent from January until the end of the campaign due to a serious knee injury.
Elsewhere, according to Sky Deutschland, Brentford are keen to sign Egyptian forward Omar Marmoush during this summer's transfer window.
The 23-year-old spent the 2021-22 season on loan at Stuttgart from Wolfsburg, finding the back of the net on three occasions in 23 appearances, in addition to registering five assists.
The report claims that Brentford have already made a 'concrete offer' for the attacker, although it is unclear whether Wolfsburg are prepared to accept the bid that has been placed on the table.
Marmoush, who is capable of playing in a number of different forward positions, only has another 12 months left to run on his contract at Wolfsburg.
Meanwhile, Leicester City are reportedly set to miss out on a deal for PSV Eindhoven midfielder Ibrahim Sangare during this summer's transfer window.
The 24-year-old has been linked with a number of clubs in the current market due to his impressive form last term, with Manchester United said to be admirers.
According to The Sun, the strongest interest is arriving from Leicester, but the midfielder is considering signing a new contract at PSV, as the Foxes have not yet made a concrete offer for his services.
Sangare was in strong form for his Dutch club during the 2021-22 campaign, scoring four times and registering four assists in 49 appearances in all competitions.