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Because low Earth orbit (LEO) satellites provide short round-trip delay, they are becoming increasingly important for providing real-time applications such as voice and video traffic to global coverage. Multicast communication that simultaneously transmits messages from a source to a group of destination is essential for future LEO satellite networks. The multicast routing problem in terrestrial networks has already been studied extensively in the past. In LEO networks, previous research on the multicast routing only considers the network cost optimization without satisfying the delay requirement. This paper proposes a distributed multicast tree algorithm with sub-optimal network cost under the delay bound in LEO satellite networks. We compare the performance of our MC algorithm when applied to the LEO satellite networks with the periodically time-variant ISL sub-network. The algorithms are judged based on the quality of MC trees it generates and its efficiency in managing the network resources. Simulation results show that our delay-constrained multicast algorithms are capable of fulfilling the delay requirements in LEO satellite networks.