The inverse kinematics of redundant manipulators require avoiding joint limits to ensure that the solutions are subject to the actual physical constraints. Current methods based on differential kinematics typically consider only local instantaneous states and cannot guarantee that the joints remain within the physical limits throughout the continuous motion. To address this issue, this paper proposes an inverse kinematics method for redundant manipulators based on model predictive control. By combining the null space parameterization of the Jacobian matrix, the proposed method effectively accounts for the future evolution of the system's kinematic states and constraints. The constraints and optimization objective functions are designed to handle joint limits, and the inverse kinematics problem is transformed into a constrained optimization problem, where redundancy is fully exploited to avoid joint limits. Furthermore, to ensure the feasibility of the optimization problem, a task scaling method is introduced to handle violations of constraints by the end-effector velocity. Simulation experiments with a 7-DOF redundant manipulator demonstrate that, compared with benchmark methods, the proposed method can predict and avoid potential joint limit violations while accurately tracking the target trajectory of the end-effector.