Multiple failure modes of gas compressor stations have not been modeled into risk evaluation of electricity-gas integrated energy system (EGIES). This paper presents a method to incorporate multiple failure modes of compressor stations with bypass systems in the EGIES risk evaluation. Three outage models representing multi-states and multi-failure-modes of compressor stations and a bi-level Monte Carlo sampling algorithm for the models are developed. A novel network model of EGIES considering compressor stations with multiple failure modes is presented. An EGIES with a modified RBTS and an 11-node gas system is used to demonstrate the effectiveness of the proposed method and models. Results indicate that compressor stations have both positive and negative significant impacts on the risk of EGIES. Bypass systems for compressor stations can effectively offset the negative impacts. Ignoring compressor stations and their station failures in the risk evaluation of EGIES may result in an inaccurate estimation of the system risk levels and even lead to a misleading conclusion in system planning.