Hydraulic structures are typically subjected to long-term hydraulic loading, and concrete—the main material of structures—may suffer from cracking damage and seepage failure, which can threaten the safety of hydraulic structures. In order to assess the safety of hydraulic concrete structures and realize the accurate analysis of the whole failure process of hydraulic concrete structures under the coupling effect of seepage and stress, it is vital to comprehend the variation law of concrete permeability coefficients under complex stress states. In this paper, several concrete samples were prepared, designed for loading conditions of confining pressures and seepage pressures in the first stage, and axial pressures in the later stage, to carry out the permeability experiment of concrete materials under multi-axial loading, followed by the relationships between the permeability coefficients and axial strain, and the confining and seepage pressures were revealed accordingly. In addition, during the application of axial pressure, the whole process of seepage–stress coupling was divided into four stages, describing the permeability variation law of each stage and analyzing the causes of its formation. The exponential relationship between the permeability coefficient and volume strain was established, which can serve as a scientific basis for the determination of permeability coefficients in the analysis of the whole failure process of concrete seepage–stress coupling. Finally, this relationship formula was applied to numerical simulation to verify the applicability of the above experimental results in the numerical simulation analysis of concrete seepage–stress coupling.