The salt cavern compressed air energy storage (CAES) reservoir undergoes periodic cyclic gas injection and production, which induce coupled fatigue-creep effects on the surrounding rock during operation. In this work, a new constitutive model was developed for salt rocks, and it uses the traditional Norton model to describe the hardening degree with state variables. The fatigue-creep mechanical properties were investigated, the model was validated, and the model parameters were analyzed using triaxial continuous and interval fatigue tests. The findings were as followed: (1) Increasing the time interval will result in a higher residual strain and a shorter fatigue life; the confining pressures increases the compressive strength of salt rock. (2) The model only needs to adjust two parameters to accurately fit the fatigue-creep deformations behavior of salt rock, and it effectively characterizes the relationship between deformation and stress routes. (3) In the model, parameters m and k play a role through state variables. Parameter m affects the magnitude of the overall deformation and the length of the deceleration stage. Parameter k controls the bending of the curve, and has a more sensitive influence on the model. (4) Compared with that of the continuous fatigue model, the floating range of parameters m and k in the interval fatigue model is less than 10%. The model can adapt to the influence of time interval on rock deformation and more accurately predict the deformations of the salt rock surrounding the salt cavern reservoir.