Cleverley was replaced by Paul Scholes just before the break after the injury flared up and Sir Alex revealed afterwards that the initial prognosis does not look good.
"We'll assess Tom in the morning," the boss told reporters at his post-match press conferece.
"He felt his calf a little but it's not good news. We'll see what he's like in the morning."
There was better news on Shinji Kagawa, however, who has not featured since 23 October at home to Braga because of a knee injury.
"Shinji is now back and is doing some running," confirmed Sir Alex. "I don't know if he'll make Sunday's game [against Manchester City]. It might be a bit early for him.
"At least he's back training and he'll hopefully join in the football part next Monday."