Aimed at the problems that the transmission bandwidth in low earth orbit (LEO) satellite networks is difficult to support the forwarding of large data volumes and make the links periodically disconnecting and reconnecting according to the needs, a satellite network multi-path routing optimization strategy is designed based on link state awareness. The dynamic topology of LEO satellites and the state of inter-satellite links are modeled, and an optimization of multi-path selection is constructed. In order to enhance the efficiency of solving the optimization problem, a multi-path selection algorithm is devised based on state monitoring and path prediction. The algorithm is to utilize time slot partitioning of the topology for mitigating its dynamism and reducing complexity by screening inter-satellite links based on monitored states, and predicting link availability based on ephemeris data to avoid massive packet loss caused by link interruptions. On the computation phase, in comprehensive consideration of real-time link states such as transmission delay, transmission bandwidth, and transmission success rate, the optimal multiple paths are selected according to the load variations to increase the throughput of satellite networks and reduce transmission delay. The simulation results demonstrate that compared to the traditional routing algorithms such as contact graph routing (CGR), shortest path first (SPF), and equal-const multi-path (ECMP), when the load is 8 Gbps, the proposed scheme in the aspects of transmission delays, is 16.9%, 11.4%, and 7.1% lower than the three algorithms respectively, and in terms of network throughput, is 34.4%, 26.9%, and 15.6% higher than the three algorithms respectively. The transmission success rate is 15.3%, 9.6%, and 5.6% higher than that of the three algorithms, respectively.