Stability of the magnetorheological rotational flow in presence of magnetic excitation in the axial direction is examined. The Galerkin projection method is used to derive a low-order dynamical system from the conservation of mass and momentum equations while mixed boundary conditions are assumed. In absence of magnetic excitation, the base flow loses its radial flow stability to the vortex structure at a critical Taylor number. The emergence of the vortices corresponds to the onset of a supercritical bifurcation. The Taylor vortices, in turn, lose their stability as the Taylor number reaches a second critical number corresponding to the onset of a Hopf bifurcation. The axial magnetic field turns out to be a controlling parameter as it alters the critical points throughout the bifurcation diagram.