This paper proposes and implements a novel asymptotic expansion approach for pricing discretely monitored American options and approximating the optimal early exercise boundary, under a generic class of models incorporating both stochastic volatility and jumps. The price and the critical value are expanded up to any arbitrary order around those under a simple constant volatility jump-diffusion model of Merton (1976). The expansion terms are then obtained by exactly solving some backward inductions via Fourier transforms, where a substantial extension of the Feng-Linetsky Hilbert transform method plays an important role. The efficiency of our method is illustrated through some representative examples.