Pogba picked up the injury in the early stages of the Champions League win over Basel, and was replaced by Marouane Fellaini after 19 minutes.
He is set to miss league games against Everton, Southampton and Crystal Palace, Wednesday's EFL Cup tie with Burton and the Champions League encounter with CSKA Moscow on 27 September.
The untimely injury has left Pogba in a race against time to be fit for United's crunch clash against Liverpool at Anfield on 14 October.
Pogba has formed a potent partnership with new signing Nemanja Matic, starting every game for United this season, scoring twice.
In tandem with the Serbian, Pogba has provided the ammunition for red-hot strikers Marcus Rashford and Romelu Lukaku and his inspirational performances have led the charge in United's blistering start to the season.
The question of who should come in for him in the engine room is a headache for boss Jose Mourinho to deal with.
The Reds are blessed with impressive strength in depth in every position with Fellaini, Michael Carrick and Ander Herrera waiting in the midfield wings.
Fellaini - hailed as a key cog in this United side by his manager - is in superb form himself and seems the first choice go-to option to fill the void left by the absent £89m man.
He was man-of-the match against the Swiss champions despite starting on the bench, and has become increasingly influential for the Reds.
After his impressive performance in a key game, Fellaini could be in line for a start against his former side at Old Trafford on Sunday.
Carrick has yet to feature this season but could get his chance with a number of fringe players set for a run out in the Carabao Cup third-round match against Burton on Wednesday.
A long-term run in the side alongside Matic seems incompatible, with both players defensively sound but limited going forward - which could potentially make us unbalanced.
However, Carrick provides a calming presence off the field as much as he does on it, with his ability to bring the best out of those around him still unrivalled.
Herrera has had to be patient in the first few weeks of the season, having only started in the UEFA Super Cup against Real and at Stoke City last time out in the league.
He is the most forward-thinking of the trio and exudes and positivity so, despite being on the periphery so far, he's more than capable of finding the form that won him the Sir Matt Busby Player of the Year award last term.