Asensio has emerged as one of Europe’s hottest prospects and scored a wonder goal against Valencia on Sunday.
But news outlet Diario Gol suggest Jose Mourinho submitted a cheeky bid for the Spaniard while making enquiries for Gareth Bale.
Real, however, laughed off the bid.
Asensio currently has a tantalising £74m release clause in his current deal.
But Real want to raise his buyout to a gargantuan £552m to ward off suitors.
Zinedine Zidane views the Spaniard as indispensable and integral to his long-term project at the Bernabeu.
Diario Gol say Asensio will be given more first-team opportunities this term.
Last season he was predominately deployed from the bench and struggled to displace Gareth Bale from the 1st XI.
Reports over the weekend suggested Arsenal were in the hunt for his signature.
The Gunners are facing a mini exodus over the next 24 hours and desperately need reinforcements.
Kieran Gibbs has already left the club for West Brom.
Meanwhile Alex Oxlade-Chamberlain and Alexis Sanchez could also be packing their bags.
Oxlade-Chamberlain is Liverpool bound and Sanchez remains a top target for Manchester City.
