Finite-control-set model predictive control (FCS-MPC) has many advantages in electric drive control systems but needs the accurate knowledge of the system parameters. The performance of the FCS-MPC will be deteriorated under parameter mismatches. This study proposes an adaptive FCS-MPC current control method for interior permanent magnet synchronous machine (IPMSM) drives subject to the inductance variations. The inductances are identified online by an adaptive observer with a recursive algorithm, which is inherently incorporated into the FCS-MPC control process to reduce the additional computational cost. Compensation methods are also proposed to improve the identification accuracy. The simulation and experimental results validate that, the IPMSM current control performance, speed-extension capability and drive efficiency are all improved by the proposed method.