The 21-year-old winger enjoyed a breakthrough campaign at the Bernabeu last term, scoring 10 goals in 38 appearances for the Champions League winners.
Despite being one of the hottest talents in Spanish football, he faces a battle to win a place in Zinedine Zidane’s star-studded team.
According to Don Balon, Manchester United and Arsenal are both willing to launch big offers for Asensio.
Juventus are also keen and will firm up their interest if they lose Paulo Dybala to Barcelona.
But all three clubs are likely to be disappointed because Real president Florentino Perez is unwilling to sell Asensio.
Don Balon report, however, that the Spain international will demand improved terms if the club want to extend his contract.
It was reported in May by Don Balon that Asensio had personally snubbed an offer from United.
German champions Bayern Munich and French giants Paris St-Germain also reportedly tabled proposals.