The coherence vortices in a new class of partially coherent beams with separable phase recently proposed by Bogatyryova et al. are studied in detail. The new beams are generated by taking an incoherent superposition of Laguerre-Gaussian (LG) modes with equal azimuthal index. It is shown that the mode indices and weight factors of superposed LG modes, as well as the reference choice, affect the position where the circular edge dislocation takes place, and result in the vortex disappearance or the appearance of more than one coherence vortices. Furthermore, there also exist coherence vortices in partially coherent beams with non-separable phase consisting of an incoherent superposition of LG modes.