Onana's move had been on the cards since the club acquired Senne Lammens from Royal Antwerp on last week's transfer deadline day.
Lammens' £18.2 million signing left United with four goalkeepers, and while the club said they were fine carrying that many, the struggling Onana's future had already been in doubt.
Turkey international Altay Bayindir started all three of United's Premier League games this season, while the 29-year-old Onana has only played once, in United's shocking second-round League Cup loss to fourth-tier Grimsby Town in a penalty shootout. Onana was at fault for Grimsby's two goals.
Onana was signed from Inter Milan for £47.2 million in 2023, and his inconsistency has made him a target for scathing criticism ever since.
He was United's number one last season, but a hamstring injury kept him out of the club's pre-season games.
Turkey's transfer window does not close until Friday, and there is anticipation that Onana, who is on international duty with Cameroon, could make his debut against Fenerbahce on Sunday.
Tom Heaton is United's fourth goalkeeper.