Jose Mourinho is already weighing up targets for next summer with the Red Devils hopeful of landing another marquee signing and Madrid star Asensio is top of his wishlist.
Spanish publication Don Balon claim United’s offer to bring him to Old Trafford has already reached the midfielder.
United are prepared to break the bank for his signature and are willing to pay around €200m (£177m).
Mourinho hopes a proposal of a five-year contract, that would see Asensio pocket £24m a year, is enough to convince the 21-year-old to leave the Bernabeu.
The Portuguese boss has plans to build a team capable of winning the Champions League and believes Asensio is key to his aims.
The Spaniard is widely regarded as one of Europe’s most prodigious talents with pundits already tipping him as a future Ballon d’Or winner -despite his team-mate Isco trying to calm the hype around the player.
“He has great potential and a massive future in the game, he has a prodigious left foot, one which practically no other player has,” he told Movistar+.
“We just have to let him keep improving with calmness and not allow his head to be filled with little birds.
“People are talking about how they think he is going to win the Ballon d’Or but we don’t talk about that and don’t think about winning that.
“All that we can do is work hard and try to get better every day.”
However, Asensio is under contract with Madrid until 2023 and Los Blancos are unlikely to want to part with their prized asset anytime soon.