For the trajectory tracking control problem of azimuth stern drive (ASD) tug, it is proposed to construct a three-degree-of-freedom motion data driving model of the tug through gated recurrent unit (GRU) neural network, and construct a model predictive control (MPC) trajectory tracking controller based on the GRU model, to overcome the restriction of the traditional control method on the precise system mechanism model. This controller regulates the tug speed and heading by adjusting the left and right rudder angles without changing the tug propeller speed, and the effectiveness of the proposed scheme is verified by simulation experiments. The model accuracy is good under noise interference. By comparing the control performance under different prediction step sizes, the influence of prediction step size on the control effect and solution time is explored. Due to the increase in the complexity of the optimization solution, when the prediction step size increases, the increase in the control accuracy leads to an increase in the solution time at the same time. This study provides new ideas for the precise trajectory tracking control of tugs, and also provides valuable references for similar nonlinear system control studies.