Varane limped off during the first half of the 2-2 draw with Atalanta on Tuesday. After undergoing scans in Manchester on Wednesday, the 28-year-old is set for a spell on the sideline and will miss Manchester City's visit to Old Trafford on Saturday.
A statement issued by the club on Wednesday read: "Manchester United can provide further information on the extent of the injury suffered by Raphael Varane during Tuesday's 2-2 draw with Atalanta in the Champions League.
"The club confirms that Raphael sustained a hamstring injury which is expected to keep him out of action for around a month. We'd like to wish Raphael a speedy recovery."
Varane has only just returned after recovering from a groin injury picked up while playing for France.
Lindelof picked up a knock in training on Monday, but the injury has been described as minor.
If manager Ole Gunnar Solskjaer decides to stick with his 3-4-2-1 system against City, he could pick Harry Maguire, Eric Bailly and Lindelof in a back three. Solskjaer has also picked Luke Shaw as a left-sided centre-back with Alex Telles filling in as the left wing-back.