Water ingress and air ingress accidents are the types of accidents that require special consideration for high temperature gas-cooled reactors (HTRs) due to the possible graphite corrosion phenomenon. In these accidents, air or steam can flow into the reactor core, corrode the graphite reflector and fuel elements matrix, which may reduce the structural strength of the graphite core components and the ability of retaining fission products inside the fuel elements. There is a strong chemistry-flow–heat transfer nonlinear coupling phenomenon in both scenarios during most of the concerned period, such that the accurate and efficient solution of this multi-physics coupling system is the first but crucial step to analyze these water ingress and air ingress accidents. In this work, the efficient Newton–Krylov (NK) method is utilized to solve this nonlinear coupling system in HTRs due to its excellent convergence rate and numerical stability. Moreover, a bounded NK method is proposed to ensure its physical property for the intermediate iterative solution. Additionally, a nonlinear elimination strategy is adopted for the intermediate variable in the mass transfer model near-wall gas concentration to reduce the scale of unknowns. The newly developed chemistry-flow–heat transfer coupling code is validated by code-to-code comparison with HTR safety analysis code TINTE, as well as comparison with NACOK (Naturzug im Core mit Korrosion) experimental data. It shows that the results of newly developed coupling code agree well with TINTE and the NACOK experimental data. The performance of the NK method is also evaluated, which showed a computational efficiency more than 8 times higher than that of the traditional Picard method, preliminarily demonstrating the high efficiency of the NK method for the chemistry-flow–heat transfer coupling issue.