In general, a magnon does not interact directly with an atom. Here we present a scheme to generate coherent coupling between two magnons and a superconducting artificial atom via exchange of virtual photons in dispersive cavities. By probing into the induced magnon-atom interactions, we identify the two-magnon process, which causes almost perfect two-mode squeezing and entanglement of the two magnons for specific conditions. The quantum entanglement of the magnon modes in two massive ferrimagnets survives at the steady state and represents genuinely macroscopic quantum state. Our work offers a...