Real Madrid forward Asensio has been linked with a move away from the Bernabeu after struggling for game time under Zinedine Zidane.
The Spain international started the campaign in fine fettle with goals against Barcelona in both legs of the Spanish Super Cup propelling him into the limelight.
However, the 21-year-old has fallen out of favour with Zinedine Zidane often preferring to pick the likes of Cristiano Ronaldo, Gareth Bale and Karim Benzema instead.
And Spanish website Diario Gol claim Real Madrid have been rocked after Asensio told them he has an offer to join United when the transfer window reopens next month.
United are set to splash the cash in the winter market with Jose Mourinho desperate for attacking reinforcements.
Despite boasting some of the best players in world football, the Red Devils are likely to miss out on the Premier League title to Manchester City.
The form of Romelu Lukaku and Henrikh Mkhitaryan has been called into question, while Marcus Rashford and Anthony Martial appear ill-suited to play together.
And Diario Gol claim Mourinho wants Asensio to boost his side’s slim chances of winning the title.
Real president Florentino Perez is now supposedly concerned one of his team’s best young players may quit for a new challenge elsewhere.
Zidane, however, is said to think differently.
The Frenchman is reportedly open to letting Asensio depart, and has conveyed his feelings on the matter to the board.
As well as United, Chelsea have also emerged as potential suitors.
The Blues are also likely to spend big next month with their season yet to properly get going under Antonio Conte.