Valverde did little to dispel the rumours that the reigning La Liga champions would be interested in making an approach, although they are likely to face stiff competition for the signature of the player if he were to become available, with Juventus offering a potential route back to Turin for their former player.
As reported by Gianluca Di Marzio, Barca have offered €50 million for the Frenchman, plus centreback Yerry Mina and midfielder Andre Gomes.
Pogba was inconsistent last season and was even dropped by the United boss for a spell in the second half of the season.
It is added that Pogba was disappointed by Mourinho's public digs at his World Cup performances.
Yet it is said that renowned agent Raiola has had discussions with Barca to discuss a move - one which Pogba himself would be open to, according to the Mail.
Pogba endured a turbulent time at Old Trafford last season.
Raiola, who took a staggering £41m cut of Pogba's then world record fee when he returned to Old Trafford from Italy in 2016, sounded out some of Europe's top clubs in January and Pep Guardiola confirmed that the player had been offered to Manchester City.
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Paul Pogba is becoming unsettled at Manchester United after feeling manager Jose Mourinho unfairly criticised him this summer, sources told ESPN FC.
The situation has not been helped by Mourinho's negative comments about Pogba's success with France at the World Cup, made on several occasions during United's recent tour of the U.S., that went down very badly with the player.
"We respect the player and the teams they are on". Mourinho appeared to question Pogba's effort and heart while playing for his club, saying, "I don't think it's about us getting the best out of him [at Manchester United], it's about him giving the best he has to give".
The France worldwide has been linked to Spanish giants Barcelona following his impressive performances in winning the World Cup with his country. Because it's closed for a month, where he can only think about football. Where he's with his team on the training camp, completely isolated from the external world, where they focus just on football, where the dimensions of the game can only motivate.
Several English media outlets report that Raiola will present a bid of €110 million that Barcelona are willing to pay the "Red Devils" to acquire Pogba.
The Old Lady want the deal completed before the transfer window closes.
United are expecting a busy few days before the 5pm transfer deadline on Thursday.