Finite volume methods are developed for pricing American options under Kou jumpdiffusion model. Based on a linear finite element space, both backward Euler and CrankNicolson full discrete finite volume schemes are constructed. For the approximation of the integral term in the partial integrodifferential equation (PIDE), an easytoimplement recursion formula is employed. Then we propose the modulusbased successive overrelaxation (MSOR) method for the resulting linear complementarity problems (LCPs). The H+ matrix property of the system matrix which guarantees the convergence of the MSOR method is analyzed. Numerical experiments confirm the efficiency and robustness of the proposed methods.