Sky Italia claimed that the Red Devils rebuffed an offer of £45million with centre-back Yerry Mina and midfielder Andre Gomes also included.
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.
The former Juventus star arrived at Manchester United back in the summer of 2016 for a club record fee of €05 million and aided the club to one League Cup and a Europa League title ever since.
It was reported yesterday that Pogba's agent, Mino Raiola, has reassured Barcelona that he will get Paul Pogba to move to the Nou Camp.
While Mourinho has refused to confirm he wants striker Anthony Martial to stay at United for the new season, he has stated a desire to keep Pogba and it remains to be seen whether that stance could shift ahead of Thursday's 5pm transfer deadline in England.
"There are a lot of names that always get circulated around in the rumour mill - all of them are good players and we respect them themselves but we also respect the clubs they are now at".
"I think the World Cup is the flawless habitat for a player like him to give [their] best", Mourinho said.
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Pogba was inconsistent last season and was even dropped by the United boss for a spell in the second half of the season.
He is under contract with Manchester United until 2021.
"Why? Because it's closed for a month, where he can only think about football".
But this isn't the first time the 25-year-old's future at Old Trafford has been in doubt.
Italian newspaper, TuttoSport reports that Juventus are ready to beat Barcelona for Pogba's signature by including Pjanic.
In theory, Barcelona would have until the end of the month to sign Pogba before the transfer window closes in Spain.
But it seems United have turned down the shockingly-low bid.