Magnetic damping controls the performance and operational speed of many spintronics devices. Being a tensor quantity, the damping in magnetic thin films often shows anisotropic behavior with the magnetization orientation. Here, we have studied the anisotropy of damping in Ta/CoFeB/MgO heterostructures, deposited on thermally oxidized Si substrates, as a function of the orientation of magnetization. By performing ferromagnetic resonance (FMR) measurements based on spin pumping and inverse spin Hall effect (ISHE), we extract the damping parameter in those films and find that the anisotropy of damping contains four-fold and two-fold anisotropy terms. We infer that four-fold anisotropy originates from two-magnon scattering (TMS). By studying reference Ta/CoFeB/MgO films, deposited on LiNbO3substrates, we find that the two-fold anisotropy is correlated with in-plane magnetic anisotropy (IMA) of the films, suggesting its origin as the anisotropy in bulk spin–orbit coupling (SOC) of CoFeB film. We conclude that when IMA is very small, it’s correlation with two-fold anisotropy cannot be experimentally identified. However, as IMA increases, it starts to show a correlation with two-fold anisotropy in damping. These results will be beneficial for designing future spintronics devices.