Media reports suggest United and Inter agreed a fee late on Sunday (July 16) worth in the region of 50 million euros, with Onana agreeing terms for a contract until 2028.
Onana would re-unite with Erik Ten Hag, his former manager at Ajax.
United saw David De Gea leave as a free agent following the expiration of his contract.
Meanwhile Inter, who also lost captain and backup goalkeeper Samir Handanovic, are working on bringing in goalkeepers Anatoliy Trubin from Shakhtar Donetsk and Yann Sommer from Bayern Munich, according to Italian media reports.