The Uruguay international has been a key player for the European champions, but reports claim that he could be persuaded to accept a new challenge.
United are reportedly one of the player's admirers and it is claimed that they could pay £85 million to prise him away from Los Blancos.
The Red Devils are set for a busy summer transfer window. They could pursue at least one marquee signing for the midfield department.
Christian Eriksen is expected to leave on a free transfer. Casemiro could be offloaded as well amid interest from the Saudi Pro League.
There is also the possibility that Kobbie Mainoo could depart.
The club graduate has snubbed the latest contract offer on the table and he could be sold for around £70 million at the end of the season.
Hence, United may have the funds to afford Valverde this summer.