Onana returned to the UK on Sunday night after withdrawing from the Cameroon squad. He picked up a lower back injury during a 3-0 win over Mauritius in World Cup qualifying on Friday and will not play against Libya on Tuesday.
The goalkeeper will undergo medical tests with United staff this week before a decision is made about whether he is available to face Everton.
A source said Onana is hopeful of being fit for the game at Goodison Park because the injury is not thought to be a muscle strain, which would take longer to heal.
If Onana isn't passed fit to face Everton, United will give a debut to summer signing Altay Bayindir. The 25-year-old arrived in a £4.3 million ($5.4m) deal from Fenerbahce but is yet to feature his season.
Luke Shaw and Tyrell Malacia are nearing comebacks after long spells on the sidelines but Rasmus Højlund and Christian Eriksen are out after picking up injuries in the 1-0 win over Luton Town before the international break.