McCarthy said Monday it will take "a couple of weeks" to determine the severity of Clifton's injury.
Clifton was bowled over by Falcons defensive end Ray Edwards in the second quarter of Sunday night's game in Atlanta. He remained on the turf, clutching his right leg, and had to be helped to the sideline and then was carted to the locker room.
Clifton did not return, but the Packers rallied to beat the Falcons 25-14.
Green Bay was already missing right tackle Bryan Bulaga, who missed his second straight game with a knee injury.
Information from The Associated Press contributed to this report.