United began their search for a short-term appointment to take the team until the end of the season on Sunday after dismissing Ole Gunnar Solskjaer. Valverde, who led Barca to LaLiga titles in 2018 and 2019 before leaving in January 2020, is one of the names in the frame along with former Borussia Dortmund manager Lucien Favre and ex-RB Leipzig coach Ralf Rangnick.
Rangnick, now working at Lokomotiv Moscow, has privately distanced himself from the job, but initial talks have been held with Valverde.
After Solskjaer's departure, Michael Carrick was put in charge temporarily, and despite his winning his first game 2-0 against Villarreal on Tuesday, sources have told ESPN that the club remain committed to bringing in someone else between now and the summer.
Sources have told ESPN that Mauricio Pochettino is United's top target as the club's next permanent appointment, but the former Tottenham Hotspur boss is under contract with Paris Saint-Germain until 2023 after signing an extension in the summer.
Pochettino, according to sources, would be interested in taking over at Old Trafford. The Argentine is in Manchester for PSG's Champions League clash with Manchester City at the Etihad Stadium on Wednesday.
And asked about the vacancy at Old Trafford at his prematch news conference, Pochettino remained tight-lipped. He said: "I am very focused on football. I am not a kid, I'm not a child; I spent my whole life playing football and now 12 years being a coach, we are in a business and the rumours are there.
"I completely understand what is going on -- sometimes it is a positive or sometimes negative, but this kind of thing cannot distract me.
"What another club is doing's not my business, and I am not going to make the mistake to talk. Things get misunderstood and out of context."