In this article, an improved multistep finite control set model predictive current control (FCS-MPCC) based on adaptive integral extended state observer (ESO) is proposed for permanent magnet synchronous motor. The improved multistep FCS-MPCC based on sector is introduced in the current loop. The elements of the voltage vector set are reduced with different sector division method, so it could reduce the computational burden to some extent. Meanwhile, considering that the high gain ESO will obtain faster convergence, better tracking accuracy in theory and the system disturbance rejection can be enhanced, but the noise suppression performance will become poor. Thus, the adaptive extended state observer (AESO) is proposed to balance the disturbance rejection and noise suppression. When system is subject to disturbance, the adaptive gain will increase to enhance the system disturbance rejection and become small to improve the noise suppression in steady state. However, disturbed by time-varying disturbance, the small gain in the steady state will lead to poor steady-state tracking accuracy. To improve the steady-state tracking accuracy, AIESO is proposed by adding the integral term into AESO. Finally, in order to avoid the sector misjudgment caused by parameter mismatch, the strategy based on ESO is used for disturbance compensation. However the selection of inductance is closed related to the estimated burden of the observer. Thus, the inductance parameter estimation method is proposed to update the initial inductance value to reduce the estimated burden, which can help to suppress the parameter mismatches with a smaller estimated burden of the observer. The simulation and experimental results verify the effectiveness of the proposed method.