In this paper, we have derived the analytical formulae for the cross-spectral densities of partially coherent Gaussian vortex beams propagating in a gradient-index (GRIN) fiber. In numerical analysis, the variations of the intensity and the phase distributions are demonstrated to illustrate the change in singularities within a GRIN fiber. It turns out that the beam intensity and phase distribution change periodically in the propagation process. The partially coherent Gaussian vortex beams do not typically possess the center intensity zero in the focal plane, which usually called 'hidden' singularities in intensities detection. We demonstrated the phase singularities more clearly by the phase distribution, one finds that the phase vortex of a partially coherent beam will crack near the focus, and opposite topological charge will be generated, we attribute to the wave-front decomposition and reconstruction of the vortex beams by the GRIN fiber. Our results show that the change in phase singularities not only affected by the GRIN fiber, but also by the initial coherence of the beam source, and high initial coherence will be more conducive to maintaining the phase singularities in the propagation. Our results may find applications in singular optics, wave-front reconstruction and optical fiber communications.