In this study, a multicast routing protocol for hyperentangled relay quantum networks is proposed to solve the problem of path selection, the establishment of multicast communication in quantum networks, and improve the communication performance of quantum multicast networks. The networks use a diamond structure to ensure the fidelity of quantum clones in multicast communication. The single-photon polarization-space mode quantum state was cloned using a quantum cloning machine, and an optimization algorithm based on the Steiner tree was used to generate a multicast tree by considering the number of hyperentanglement resources, hops, and fidelity of the cloned state. A multicast quantum channel between remote users was established based on simultaneous hyperentanglement swapping after selecting the routing. Theoretical analysis and simulation results show that in multicast communication of an increase in destination nodes, the routing protocol based on a hyperentangled relay can obtain a multicast tree with high fidelity. When the number of relay nodes increases, the delay in establishing a quantum channel using simultaneous hyperentangled swapping is lower than that of traditional sequential entangled swapping, and the transmission rate of a quantum state is faster. Therefore, the routing protocol of quantum multicast networks based on hyperentangled relays has the advantages of high fidelity and low-multicast communication delay.