As reported by Gianluca Di Marzio, Barca have offered €50 million for the Frenchman, plus centreback Yerry Mina and midfielder Andre Gomes.
Barcelona are believed to be interested in the 25-year-old, who won the World Cup with France this summer, while Juve are also been linked with their former player.
Barcelona will be hard pressed to convince United to part with Pogba with the English transfer window set to shut on Thursday.
Paul Pogba is becoming unsettled at Manchester United after feeling manager Jose Mourinho unfairly criticised him this summer, sources told ESPN FC.More news: Willian Confirms FC Barcelona Offer, Chelsea FC Stay
More news: Bayern beat Man United in final pre-season test
More news: Pope declares death penalty inadmissible, changing Church's stance
Jose Mourinho tells ESPN FC that the World Cup was the ideal habitat for Paul Pogba as he could 'only think about football'. With that out of the way he's coming good though we all know he can do even better.
United are aware that it could be a ploy by Pogba's representative to secure him a new contract on the same terms as £500,000-a-week top earner Alexis Sanchez.
'The players are in a closed camp, isolated from the external world, from commercial compromises, and every possible influence'.
Barca can still look to poach Pogba until the end of August, with the La Liga deadline not until August 31, although United would be unable to bring in a replacement - virtually dismissing any possibility of a deal. Bayern want £50m for the former Manchester City defender, but United are unwilling to pay more than £40m for a player who turns 30 next month.
The Old Lady want the deal completed before the transfer window closes.