Superfluid He-3-B possesses three locally stable vortices known as a normal-core vortex (o vortex), an A-phasecore vortex (v vortex), and a double-core vortex (d vortex). In this work, we study the effects of a magnetic field (less than or similar to 0.1 T) parallel or perpendicular to the vortex axis on these structures by solving the two-dimensional Ginzburg-Landau equation for two different sets of strong coupling correction. The energies of the v and d vortices have nontrivial dependence on the magnetic field. As the parallel magnetic field increases, the v vortex is energetically unstable even at high pressures and the d vortex becomes energetically most stable at all possible pressures. In perpendicular magnetic field, the energy of the v vortex is lower than that of the d vortex in the high pressure regime. In addition, the orientation of the double cores in the d vortex prefers to be parallel to the magnetic field at low pressures, while the d vortex with the double cores perpendicular to the magnetic field is allowed to continuously deform into the v vortex with increasing pressure.