The goalkeeper is expected to sign a five-year contract with the club, with a deal close to €50 million (£43 million) on the verge of being agreed upon, including additional add-ons of approximately £5 million.
Ten Hag had made it a priority to sign a new number-one goalkeeper for Manchester United this summer, and the departure of David de Gea ensures that the club will have a new first-choice goalkeeper for the first time in 12 years.
Onana has already discussed with Ten Hag how his style of play will benefit United, allowing them to play with a higher defensive line, dominate more territory, and increase possession by playing out from the back.
Reports from Italian publications Gazzetta dello Sport and Corriere dello Sport suggest that Onana received reassurances from Ten Hag about the progress of the deal, with negotiations between the two clubs ongoing to finalize the finer details.
It is anticipated that Onana will travel to Manchester on Sunday or Monday to undergo medical tests and complete the transfer, enabling him to join United on their pre-season tour starting next week.