In this note, we treat a two-player zero-sum Dynkin game on two stocks driven by geometric Le'vy processes, for a given terminal reward cost. Explicit forms for the optimal stopping times and the value of the game are both sought for, under certain conditions. The present note extends a recent result of the author to include a wider class of diffusion processes with jumps. The main result is derived following a decomposition of a stopping game into two standard optimal stopping problems which is due to Yasuda for a standard Brownian motion.