We study the Euler equations of compressible isentropic gas dynamics with damping and spherical symmetry. For spherically symmetric flow, the global existence of the weak entropy $L^{\infty}$ solutions with damping isn’t still obtained. In this paper, we prove the existence of global solutions with small $L^{\infty}$ data. We construct the approximate solutions by using modified Godunov scheme. A $L^{\infty}$ bound for the approximate solutions can be obtained with the aid of the presence of the damping term.