While using the spiral phase plate (SPP) to produce Gaussian vortex beams, the center of the beam and the center of the SPP should be superposed. However, the center of the beam is always away from the center of the SPP to some extent in practice. The real output light from the phase plate are off-center Gaussian vortex beams. For conventional vortex beams, the core and the centroide are the same point and always located at the center during propagation. The off-center Gaussian vortex beam is different from the conventional vortex beams and its propagation is investigated in this paper. Research result shows that the core and the centroid of the off-center vortex beam are separate and moving during propagation. The directions of movement of the core and the centroid are decided by the sign of the topological charge. The movement distance of the core has no connection with the magnitude of the topological charge. However, the movement distance of the centroid is related to the magnitude of the topological charge.