The placement of a magnetic monopole into an electrically neutral chiral plasma with a nonzero axial density results in an electric polarization of the matter. The electric current produced by the chiral magnetic effect is balanced by charge diffusion and Ohmic dissipation, which generates a nontrivial charge distribution. In turn, the latter induces a separation of chiralities along the magnetic field of the monopole due to the chiral separation effect. We find the stationary states of such a system, with vanishing total electric current and stationary axial current balanced by the chiral anomaly. In this solution, the monopole becomes "dressed"with an electric charge that is proportional to the averaged chiral density of the matter - forming a chiral dyon. The interplay between the chiral effects on the one hand, and presence of magnetic field of the monopole on the other, may affect the evolution of the monopole density in the early Universe, contribute to the process of baryogenesis, and can also be instrumental for detection of relic monopoles using chiral materials.