Finite control set model predictive control(FCS-MPC) is a novel high-performance control algorithm for a three-phase voltage source PWM rectifier. But the algorithm's switching frequency is not fixed and the sampling frequency is too high. A finite switching sequence model predictive current control(FSS-MPCC) of a three-phase voltage PWM rectifier is proposed to solve this problem. The proposed algorithm selects 12 switching sequences to from the system control set, and than builds the objective function which containing the deviation of the current and switching frequency. The action time of each switching sequence is calculated based on the deadbeat fashion. The optimal switching sequence which can minimize the objective function and make the action time greater than zero is chosen from the system control set by using the online optimization. The simulation and experimental results show that the proposed FSS-MPCC can ensure a three-phase voltage source PWM rectifier's good dynamic and static performance with the fixed switching frequency and low sampling frequency.