We study the fundamental group of an open n-manifold of nonnegative Ricci curvature with some additional condition on the Riemannian universal cover. We show that if the universal cover satisfies certain geometric stability condition at infinity, the \pi_1(M) is finitely generated and contains an abelian subgroup of finite index. This can be applied to the case that the universal cover has a unique tangent cone at infinity as a metric cone or the case that the universal cover has Euclidean volume growth of constant 1-\epsilon(n).