The fission barriers as well as different fission paths of the $^{70}$Se nucleus have been microscopically investigated within a constrained Skyrme-Hartree-Fock plus BCS approach using the SkM* effective interaction. The effects of intrinsic reflection asymmetric deformations have been taken into account while axial symmetry around the fission direction has been assumed. The two-body part of the center of mass correction has been included in a perturbative way and has been found to weakly contribute to the calculated barrier heights. Upper limits of the latter have been evaluated as the heights with respect to the ground state energy of the lowest saddle points obtained along continuous paths in a reasonably dimensioned collective space connecting one-body shaped and two-body shaped configurations. They have been found somewhat higher than what is expected from other theoretical studies.