Varane had to be replaced by Dayot Upamecano just before half-time in Les Bleus' 2-1 victory over Spain.
The centre-back went to ground off the ball and was quickly withdrawn from the match, with United now confirming he sustained a groin injury.
A brief statement read: "Raphael Varane sustained a groin muscle injury in the Nations League final and has commenced rehabilitation at the club.
"He will be out for a few weeks."
It comes as a massive blow to United and manager Ole Gunnar Solskjaer ahead of a potentially pivotal few weeks.
While it is unclear exactly how many games Varane will miss, the central defender will be absent for a chunk of what looks set to be a gruelling run of matches.
United face Leicester City, Atalanta twice, Liverpool, Tottenham and Manchester City all in the next four weeks.
After a rather erratic start to 2021-22 consisting of just five wins from 10 matches across all competitions, Solskjaer heads into the upcoming run under pressure.
To make matters worse, there was already uncertainty over the condition of captain – and Varane's regular centre-back partner – Harry Maguire, who missed the clashes with Villarreal and Everton due to a calf injury before the international break that prevented him from linking up with England.
Whether Maguire will be fit to face former club Leicester at the weekend is yet to be determined, meaning the rather haphazard Eric Bailly and the dislodged Victor Lindelof could form an unfamiliar partnership at the back.
Although United has come in for criticism so far this term, Varane's early form has largely been considered positive and he has performed an important function in getting the team on the front foot, with his 7.6 passes into the final third per 90 minutes the most among the club's defenders.