The Cameroon goalkeeper is expected to officially move to Old Trafford on Thursday having reportedly passed a medical and agreed a five-year contract that includes the option for a further 12 months.
Onana was pictured on social media on Thursday at Manchester Airport waiting to board a flight to New York where he will meet up with his new United team-mates on their pre-season tour of the United States.
United and Inter Milan had been in protracted negotiations over the transfer of the 27-year-old for some time, before finally agreeing a £47.2million deal earlier this week.
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An official announcement of the deal is now expected soon, with Onana having already confirmed his switch to Manchester in an emotional Instagram message targeted at Inter supporters.
“I never imagined that a farewell would be so hard,” Onana wrote of saying goodbye to Inter, for whom he made 41 appearances across all competitions and helped reach the Champions League final last season as well as lifting both the Coppa Italia and Supercoppa Italiana following a free transfer from Ajax.
“A year ago I became a Nerazzurro, many people told me that this club and this city had something special. Now I can confirm it. In my first days as a Nerazzurro, I realised that this is a family and that being Interisti is a way of understanding life.
“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. I do it with great enthusiasm, but knowing that I will miss you. Eternal gratitude to all the Interisti. You make this club great.
“Maybe one day our paths will cross again, but now it’s time to face new challenges and cheer you on from afar. Players come and go, but the club always remains, and Inter is above everything and everyone.”
Onana joins United as the replacement for David de Gea, who spent 11 years as first-choice goalkeeper at Old Trafford but had received notable criticism following a number of high-profile mistakes last season despite winning the Golden Glove award for most Premier League clean sheets.
De Gea had looked set to stay at United on reduced terms this summer, only to end up leaving the club last month following the expiration of his contract and no agreement reached over an extension.