Real Madrid star Asensio is in-demand after taking the European Under-21 Championships by storm with Spain.
The youngster scored 10 goals from 38 games for the La Liga and Champions League winners last term - despite spending much of the campaign stuck behind Cristiano Ronaldo, Gareth Bale and Karim Benzema in the pecking order.
As a result, Europe's elite are beginning to circle.
Manchester United and Arsenal both want to add more firepower to their squad with Jose Mourinho and Arsene Wenger both targeting a Premier League title challenge next term.
And Spanish outlet Don Balon claim Asensio is on their radar.
The Premier League duo are supposedly willing to break the bank for Asensio, despite knowing Real will not be willing to let him go.
His suitors are willing to offer him regular game time and a stage to make a name for himself - something Madrid currently cannot.
The report also adds that Juventus could be about to join the race for the 21-year-old.
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Manchester United and Arsenal both admire Real Madrid star Marco Asensio
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Juventus could go after Marco Asensio if Paulo Dybala joins Barcelona
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Barcelona are likely to lose Neymar to Paris Saint-Germain, despite the Brazil international starring with a brace in their pre-season win over Juve.
As a result, they could use the funds gathered from any potential sale to raid the Serie A giants for Paulo Dybala.
And Juventus could then consequently go after Asensio.