We consider the motion of N point vortices in the inviscid and incompressible flow on a sphere with fixed pole vortices. We focus on a steady configuration, called the N-ring, where the N point vortices are equally spaced along a line of latitude. Applying a Melnikov-type analysis for Hamiltonian dynamical systems with the saddle-center fixed points to the present problem, we show that the unstable manifold and the stable manifold to the periodic orbits in the neighborhood of the N-ring cross transversely, and thus there exists a chaotic orbit in the N vortex problem on sphere.