We propose a possible origin of FME linewidth anisotropy shown in mechanically polished YIG and BCVIG single crystals. The stresses left on the surface of the spheres of these crystals couple with the magnetization through the magnetostriction effect, and thus give rise to an additional linewidth, which is anisotropic due to the anisotropic magnetostrictive constant. Supposing that the damaged layer is mainly composed of high density dislocations, a relation between the FME linewidth anisotropy and the magnetostriction constants is deduced based on micromagnetism theory and the two-magnon scattering process. From the measured linewidth anisotropy of YIG, we evaluated the dislocation density of the surface layer to be 5.4×1010/cm2.This relaxation process shows that in order to further reduce the linewidth of an ultra-low linewidth polycrystalline material, suppressing the magnetostriction constant may play an important role.