Sky Sports said United will pay €51 million (A$83 million) for the 27-year-old, plus four million (A$6.5 m) in add-ons.
“Manchester United has a long history of incredible goalkeepers, and I will now give everything to create my own legacy in the coming years,” Onana said in a statement.
He said goodbye to Inter fans earlier on Thursday.
“I fell in love with San Siro, la Curva and the city. Now the circumstances are right for me to start a new journey in Manchester,” Onana said in a statement.
“I do it with great enthusiasm, but knowing that I will miss you. Eternal gratitude to all the ‘Interisti’. You make this club great.”
United have been looking for a new first-choice keeper after Spaniard David De Gea’s contract expired in June, bringing an end to his 12-season tenure with the Old Trafford side.
In the UK, Glasgow newspaper The Daily Record has reported that the Spain international has rejected the chance to move to Inter and replace Onana in a team swap.
The Milan club apparently offered De Gea €5 million a year, but he is believed to be holding out for closer to €10m.
Onana joined Inter in July 2022 from Ajax Amsterdam and quickly became first-choice. He was a central figure in a campaign that ended with Inter winning the Italian Cup and Super Cup. They also reached the Champions League final.
It was with Ajax in Europe’s elite club competition that Onana came to prominence as an integral part of the team that went from the early preliminary knockout rounds to the 2019 semi-finals before a dramatic elimination by Tottenham Hotspur.
United finished third in the 2022-23 Premier League campaign, securing a spot in the Champions League.
Onana looked destined to be the long-term first-choice keeper for Cameroon but decided to retire from international duty after playing just one match at the 2022 World Cup, where he was suspended for disciplinary reasons.
He won 34 caps for the national team.
AAP
