Due to the high nonlinearity and uncertainty of Autonomous Underwater Vehicles (AUVs), designing a stable nonlinear Model Predictive Control (MPC) controller is very challenging. To address this issue, this paper proposes an MPC strategy based on Gaussian Processes (GP) to improve controller stability. The proposed approach employs a control framework based on a nominal model and a learnable model. The nominal model describes the main characteristics of the AUV, thereby ensuring the stability of the control system. The learnable model is trained using a sparse GP. In the offline phase, the GP model is pre-trained using historical data and integrated into the MPC. In the online phase, new observational data is collected, added to the existing dataset, and the GP model is updated using the current dataset. The feasibility and effectiveness of the proposed GP-MPC control strategy are validated through a numerical example, demonstrating significant improvements in control accuracy and stability compared to traditional methods.