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In this letter, we propose a market-based bi-level conic optimal energy flow (OEF) model of integrated electricity and natural gas systems (IENGSs). Conic alternating current optimal power flow (ACOPF) is formulated in the upper-level model, and the generation cost of natural gas fired generation units (NGFGUs) is calculated based on natural gas locational marginal prices (NG-LMPs). The market clearing process of natural gas system is modeled in the lower-level model. The bilevel model is then transferred into a mixed-integer second-order cone programming (MISOCP) problem. Simulation results demonstrate the effectiveness of the proposed conic OEF model.