United said Pogba had signed a five-year contract but did not give the fee which British media have put at 89 million pounds ($116.4 million, 105 million euros).
That would see the 23-year-old eclipse the 85 million pounds Real Madrid paid Tottenham Hotspur for Gareth Bale in 2013.
“This is the right club for me to achieve everything I hope to in the game,” said Pogba.
The 23-year-old moved to Italy from Old Trafford on a free transfer in 2012.
“I am delighted to rejoin United. It has always been a club with a special place in my heart and I am really looking forward to working with (United manager) Jose Mourinho,” added Pogba.
Mourinho has been tasked with bringing back the glory days to the 20-time English champions after three lean years since Alex Ferguson’s retirement.
And the former Chelsea boss believes Pogba can inspire United’s revival.
“He has the chance to be at the heart of this club for the next decade and beyond,” said Mourinho.
“Paul is one of the best players in the world and will be a key part of the United team I want to build here for the future. He is quick, strong, scores goals and reads the game better than many players much older than he is.”
Pogba made just seven appearances during his previous spell at the club as a teenager, but returns as one of the world’s most famous players after winning four consecutive Serie A titles and leading France to the Euro 2016 final.
“Thanks @juventusfc and all Juventini. Goodbye means nothing, it is the time we spent together that matters,” Pogba posted on his official Twitter account.
United brought the long-running transfer saga to an end in the early hours of the morning UK time.
Pogba will become United’s fourth close-season signing after Ivorian centre-back Eric Bailly, Armenian playmaker Henrikh Mkhitaryan and Swedish striker Zlatan Ibrahimovic.
Mourinho’s reign got off to a successful start on Sunday as Ibrahimovic scored the winner in a 2-1 victory over Premier League champions Leicester City in the Community Shield.