The manufacturer reports pulling away on wet roads could see the vehicle's drivetrain experience torque peaks.
If the condition occurs, it could lead to the failure of the drive-shaft, resulting in a "loss of propulsion".
Mercedes-AMG says the fault may increase the risk of an accident.
Affected C63 vehicles were sold between 1 February 2015 and 31 July 2016.
Owners of the affected cars should contact their local Mercedes-Benz dealer to arrange an update to the vehicle's ESP control unit software and, if necessary, the suspension control unit.