The interaction between natural gas and electricity networks is becoming more significant due to the projected large penetration of renewables into the energy system to meet the emission targets. This is due to the role of gas-fired plants in providing backup to renewables as the linkage between these networks. Therefore, this paper proposes a deterministic coordinated model for the secure and optimal operation of integrated natural gas and electricity transmission networks by taking into account the N-1 contingency analysis on both networks. In order to reduce the computational burden and time, an iterative algorithm is proposed to select the critical cases and neglect other contingencies, which do not have a significant impact on the energy system. The proposed integrated mixed-integer nonlinear programming operational model is evaluated and compared to another enhanced separated model on the IEEE 24-bus and 15-node gas test systems. The results emphasize the importance and effectiveness of the proposed framework (up to 6.7% operational costs savings are achieved).