This paper presents a sensorless speed control algorithm based on Finite Control Set Model Predictive Control (FCS-MPC) for Permanent Magnet Synchronous Motor (PMSM) fed by a 3-level Neutral-Point Clamped (NPC) converter. The proposed scheme uses an anti-windup Proportional-Integral (PI) controller concept to generate the reference electromagnetic torque using the error of speed. Then, FCS-MPC uses this torque reference and other parameters such as a current limitation, neutral point voltage unbalance, and switching frequency to control the converter gate signals. Also, an Adaptive Nonsingular Fast Terminal Sliding Mode Observer (ANFTSMO) was employed to estimate rotor position precisely in positive (clockwise) and negative (counterclockwise) speed to eliminate the encoder. The proposed algorithm has fast dynamics and low steady-state error. Moreover, torque fluctuation and current distortion reduced compared with Space Vector Pulse Width Modulation (SVPWM) based speed control and Direct Predictive Speed Control (DPSC). Simulation results using MATLAB/SIMULINKÒ demonstrate the performance of the proposed scheme.