In the case of the KoBoL model with the jump process (KoBoLJ), the pricing problem of American call option is investigated in this paper. The pricing model of this kind of financial derivatives is a free boundary problem with a fractional-partial-integro-differential equation (FPIDE). In fact, it is impossible to obtain the analytical solution of the mathematical model. Hence, the mathematical model with free boundary should be changed as a fixed one and then the numerical scheme is set to solve the transformed model. In the proposed approach, we proved that the American call option values obtained by the current method are not lower than the intrinsic values of this option. Moreover the PCGNR method with the fast Fourier transform (FFT) technique was employed to handle the semi-globalness of the fractional-integro operator. The significant effects of the parameters in our model on the optimal exercise price curve ware analyzed.