According to the Spanish newspaper Nacional, the Red Devils want to sign the Uruguayan and take advantage of his fluctuating form by bringing him to Old Trafford.
Following the World Cup, the Spanish giants have witnessed a string of poor performances and outcomes, with Valverde apparently being one player who has failed to make an influence on games.
Valverde will be crucial in Europe.
Real Madrid’s return to Champions League activity in the coming weeks could provide him with an opportunity to prove himself.
They will face Liverpool in the Round of 16 on February 21, before hosting the Merseyside club at the Santiago Bernabeu on March 15.
Last season’s final pitted the clubs against each other, with Real winning the tie and claiming their 14th trophy in the competition.