In this paper, we study the optical anisotropy induced by femtosecond laser radiation in air during an optical breakdown. Using a transverse pump-probe technique, we demonstrate that this anisotropy appears in a narrow range of pump intensities, which are close to the optical breakdown threshold in air and lead to a phase shift of probe radiation, polarized collinear to the pump. The intensity range where an induced intense anisotropy occurs makes it possible to estimate the magnitude of the 5th-order Kerr nonlinear refractive index component in air.