The goalkeeper has been a top priority for the Red Devils since the departure of David de Gea.
Personal terms have already been agreed upon, with Onana set to sign a five-year contract.
The initial fee for the Cameroon international is reported to be £43 million, potentially rising to around £47 million with add-ons.
The deal is expected to be announced soon, and United will work on processing Onana’s visa to join their pre-season tour in the US.
There is a possibility that he could feature in the friendly against Arsenal, or alternatively in matches against Borussia Dortmund or Real Madrid later in the month.
Onana’s arrival could also pave the way for Dean Henderson’s departure, with Nottingham Forest showing interest in the goalkeeper after his loan spell with them last season.