Asensio is billed as one of Real Madrid's stars of the future.
However, the 21-year-old has struggled to nail down a spot in Zinedine Zidane's starting line-up.
When given the chance to impress he has taken it, scoring seven goals and registering one assist from 23 games across all competitions.
And with his contract expiring in 2022, Madrid are under no obligation to sell up.
Five most likely summer deals according to the bookies
Wed, February 1, 2017
Express Sport takes a look at the five most likely deals that could happen this summer. Odds provided by BetStars
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Griezmann to the Premier League – Man Utd 15/8, Chelsea 6/1, Man City 8/1
Man United, Chelsea and Arsenal all want to sign Marco Asensio
Marco Asensio has grown frustrated at Real Madrid
But Spanish website Diario Gol claims Asensio's lack of game time has alerted the Premier League's elite.
Manchester United were told to stump up £40million for the player this month.
Zinedine Zidane wants to keep Marco Asensio at Real Madrid
However, it is said Asensio has decided to quit the Bernabeu at the end of the season.
The youngster wants regular first-team football and knows a summer move is needed for him to fulfil his potential.
James Rodriguez and Isco are currently both above him in the pecking order.