<p>We present a scheme to generate nonreciprocal entanglement and asymmetric steering between an atomic ensemble and a magnon based on Kerr nonlinearity of magnon in an yttrium iron garnet sphere. In particular, a cavity optomagnonic system is under our consideration, where the optical cavity couples with an ensemble of <i>N</i> two-level atoms, and meanwhile nonlinearly interacts with the magnon mode via optomagnonic coupling. The results demonstrate that the steady-state macroscopic quantum correlations including magnon-atomic ensemble entanglement and Einstein–Podolsky–Rosen steering could be obtained via strongly driving the cavity mode. More importantly, tuning the direction of the static magnetic field leads to a positive or negative magnon Kerr coefficient, which leads to a corresponding shift in magnon frequency and thus induces the nonreciprocity of entanglement. Furthermore, the one-way steering between magnon and atomic ensemble is also shown via properly choosing the coupling strengths and effective Kerr parameters. Our work could have potential applications in the preparation of macroscopic quantum states and be applied to construct long-distance quantum networks.</p>