  We investigate the causality and the stability of the relativistic viscous
magneto-hydrodynamics in the framework of the Israel-Stewart (IS) second-order
theory, and also within a modified IS theory which incorporates the effect of
magnetic fields in the relaxation equations of the viscous stress. We compute
the dispersion relation by perturbing the fluid variables around their
equilibrium values. In the ideal magnetohydrodynamics limit, the linear
dispersion relation yields the well-known propagating modes: the Alfv\'en and
the magneto-sonic modes.In the presence of bulk viscous pressure, the causality
bound is found to be independent of the magnitude of the magnetic field. The
same bound also remains true, when we take the full non-linear form of the
equation using the method of characteristics. In the presence of shear viscous
pressure, the causality bound is independent of the magnitude of the magnetic
field for the two magneto-sonic modes. The causality bound for the
shear-Alfv\'en modes, however, depends both on the magnitude and the direction
of the propagation. For modified IS theory in the presence of shear viscosity,
new non-hydrodynamic modes emerge but the asymptotic causality condition is the
same as that of IS. In summary, although the magnetic field does influence the
wave propagation in the fluid, the study of the stability and asymptotic
causality conditions in the fluid rest frame shows that the fluid remains
stable and causal given that they obey certain asymptotic causality condition.
