In this paper we consider the numerical solution of the two-dimensional time-dependent partial integro-differential complementarity problem (PIDCP) that holds for the value of American-style options under the two-asset Kou jump-diffusion model. Following the method of lines, we derive an efficient numerical method for the pertinent PIDCP. Here, for the discretization of the nonlocal double integral term, we employ an extension of Toivanen’s fast algorithm in the case of the one-asset Kou jump-diffusion model. For the temporal discretization, we study a useful family of second-order diagonally implicit Runge–Kutta methods. Their adaptation to the semidiscrete two-dimensional Kou PIDCP is obtained by means of an effective iteration introduced by d’Halluin and co-workers. Numerical experiments are presented that show the proposed numerical method achieves favorable second-order convergence to the American two-asset option value as well as its Greeks Delta and Gamma.