Existing motion planning algorithms for multi-robot systems must be improved to address poor coordination and increase low real-time performance. This paper proposes a new distributed real-time motion planning method for a multi-robot system using Model Predictive Contouring Control (MPCC). MPCC allows separating the tracking accuracy and productivity, to improve productivity better than the traditional Model Predictive Control (MPC) which follows a time-dependent reference. In the proposed distributed MPCC, each robot exchanges the predicted paths of the other robots and generates the collision-free motion in a parallel manner. The proposed distributed MPCC method is tested in industrial operation scenarios in the robot simulation platform Gazebo. The simulation results show that the proposed distributed MPCC method realizes real-time multi-robot motion planning and performs better than three commonly-used planning methods (dynamic window approach, MPC, and prioritized planning).