Many power systems in the United States and elsewhere are experiencing simultaneous increases of the gas-fired and renewable portions in the generation profile. Both contributions are sufficiently clean to replace retiring generators, which are mainly coal-fired. Moreover, pairing gas and renewables is advantageous because the former is flexible enough to mitigate the exogenous fluctuations of the latter. However, the resulting strong coupling of power systems and gas transmission networks through gas-fired generators also imposes risks. In particular, excessive fuel usage by gas-fired power plants may lead to violation of gas pressure limits and gas supply shortages. To provide a simple tool for assessing these risks, we develop a computational framework that characterizes regions of generator dispatch solutions that maintain gas system feasibility. The proposed algorithmic framework is modular - built through a coordinated execution of multiple generation scenarios within power and gas simulation modules. Monotone dependence of the gas pipeline pressure on the rates of gas withdrawals allows to establish and certify regions of feasibility/infeasibility in the space of the gas injections. The framework is validated against simulations of a highly detailed benchmark gas-electricity model. We conclude the manuscript with a discussion of possible applications of the results to power system operation procedures and regulatory practices.