The 22-year-old is a key player in Zinedine Zidane's side, having worked his way into the first team over the last three seasons. Valverde still has five years left on his contract and a mammoth £640m release clause in his contract.
However, as reported by the Daily Mail who quote the Sun, United are interested in Valverde and could offer Pogba in a swap deal with the Spanish giants.
The Red Devils recently activated Pogba's one-year extension which means he is tied to the club until 2022, but the Frenchman has claimed that it is 'his dream to play for Real Madrid' one day.
The midfielder said this week while on international duty with France: "Yes, all football players would love to play for Real Madrid. It is a dream for me, why not one day?"
Valverde has made 74 appearances for Los Blancos across competitions, and five this season in the LaLiga, scoring one goal until now.