In electricity market (EM) and natural gas market (GM), increasingly interacted energy flows and economic flows call for an appropriate market coordination mechanism to improve the overall market efficiency. In this paper, two equilibrium models of coordinated electricity and natural gas markets are proposed with difference in the coupling information exchanging channels between EM and GM. After fully considering the participants, especially the supply-side integrated strategic firm (ISF) and the demand-side regional integrated energy system (RIES), this paper numerically analyzes the social welfare, market power, participants’ profits/costs and market price in two equilibrium models. In addition, the overall social welfare of coordinated markets is compared with traditional uncoordinated markets to illustrate the improvement in market efficiency. The bi-level market equilibrium models are converted into a linearized equilibrium problem with equilibrium constraints (EPEC) which is a mixed-integer linear program (MILP) and can be efficiently calculated by commercial solvers.