That's according to Marca's Carlos Carpio, who reported Los Blancos are "determined" to recruit him in the summer.
As they want to ensure the deal is completed in good time, "they have set to work well in advance."
Speculation linking De Gea with Madrid has increased of late, particularly after the Spaniard was left out of United's Premier League trip to Sunderland on April 9.
The goalkeeper came close to joining Real in the summer of 2015 only for the deal to fall through at the last minute, and shortly after, the 26-year-old signed a new deal at the club to extend his stay until 2019.
De Gea has established himself as not only one of the best players in the Premier League but also among the best in his position in world football.
Per Squawka Football, he has enjoyed another excellent season:
With hardly a vintage defence in front of him, it speaks volumes of his ability that only Tottenham Hotspur have conceded fewer than United's 24 goals this term.
He showed his shot-stopping against Manchester City on Thursday to keep out Sergio Aguero, per United's official Twitter feed:
Real are yet to adequately replace Iker Casillas in goal, with neither Keylor Navas nor Kiko Casilla convincing at the top level, so a top goalkeeper is a priority and De Gea fits that bill perfectly.
Hesham Bilal-Hafiz of Manchester Evening News believes the Spaniard is sure to leave, but United could mitigate his loss if they handle the deal well:
Indeed, while the Red Devils would still need to find an adequate replacement for De Gea, the arrival of a star player who could meet one of their other pressing needs would perhaps be better than just a large transfer fee.
If De Gea does depart, it would fall to Sergio Romero to take over until a new 'keeper could be found. The Argentinian has impressed under United manager Jose Mourinho, so they would have viable cover in the short term at least.