Pogba, 23, returns to Old Trafford four seasons after departing for Italy, with Juventus confirming the transfer fee of €105 million (£89.3 million, $116.4 million).
The fee, payable over the next two years, can increase by an additional €5 million if certain conditions are met, Juventus said.
The transfer beats the previous record fee of an estimated €100 million (£85 million) that Real Madrid paid Tottenham for Gareth Bale in 2013.
When Pogba joined Juventus in 2012, United received just £800,000 in compensation, and the Frenchman quickly established himself as a key player, helping the Bianconeri win Serie A four times and the Coppa Italia twice.
Pogba had made just seven appearances for United during his first spell with the Premier League club, all during the 2011-12 season, including three in the Premier League, three in the Capital One Cup and one in the Europa League.
He returns to United on a five-year contract, with an option to extend the deal for an additional season.
Rivals Manchester City have also confirmed the signing of John Stones from Everton on a six-year contract.
City boss Pep Guardiola had already confirmed that England centre-back Stones, 22, was one of his top summer targets.
The fee has not been disclosed but Everton said it is a club record amount.
Stones told City’s official website: “I’m absolutely delighted to sign for City and now the deal’s done I’m looking forward to the next stage in my career.
“I’ve obviously seen what’s going on here at the Etihad, it’s an ambitious club with a great manager so I can’t wait to get stuck in and help achieve the goals.
“I know it will be tough getting a place in this team but I’m determined to become the best player I can be and help us to success.
“I had a wonderful time with Everton and wish them and their fans all the best but now I’ll give my all for City.”
City’s director of football, Txiki Begiristain, added: “John is an excellent young English defender and we are all really pleased he is joining us. We have worked hard to bring him to City and we feel we are signing one of the best central defenders in Europe.”
Guardiola said: “We want to help John show his quality with us and improve on what he has already achieved.
“I like the way he plays the game and I’m looking forward to welcoming him into the squad.”
The confirmation of the deal comes after UEFA listed Stones in City’s Champions League squad for the playoff round on Tuesday morning