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The Spain international impressed immensely under Zinedine Zidane last season but struggled to earn playing time.
Asensio, 21, started just 11 La Liga games as Real claimed the title ahead of Barcelona.
And Spanish publication Dario Gol say the winger is a transfer target for a number of clubs.
They claim Liverpool, Arsenal and Juventus have approached Asensio with an offer.
Diario Gol reckon Asensio wants to remain at the Bernabeu but he is willing to leave if Zidane cannot assure him of regular game time.
Zidane and Real president Florentino Perez are desperate to retain the winger.
The pair see Asensio as a long-term replacement for Cristiano Ronaldo in Madrid.
Diario Gol add that Asensio is hoping Gareth Bale moves out of the Bernabeu so he gets more minutes under Zidane.
Bale, 28, has been heavily linked with a move away from La Liga, with Manchester United actively pursuing the former Tottenham talisman.