Assuming that soil skeleton satisfies the constitutive relation of the standard linear viscoelastic solid, the transient dynamic responses of the coupled system of saturated viscoelastic soil-elastic lining of a deep circular tunnel subjected to a blasting load are studied. First, based on the Biot model of the saturated soil and the elasticity theory of the lining, with the boundary conditions as well as the continuity conditions on the interface between the saturated viscoelastic soil and elastic lining, the analytical solutions of the displacements and stresses as well as the pore water pressure of the saturated viscoelastic soil and elastic lining in the Laplace transformed domain are derived by means of the Laplace transform and potential functions. Then, the transient dynamic responses of the coupled system in the time domain are obtained with the Crump numerical inverse Laplace transform, and the radial displacements and circumferential stresses of the soil-lining coupled system as well as the pore water pressures of the saturated soil for different soil models are analyzed numerically. It is shown that, for the soil-lining coupled system with different soil models, their dynamic behaviors are almost the same under the blasting load, whereas their vibration periods and amplitudes are obviously different. Furthermore, for the system of saturated viscoelastic soil-elastic lining, the influences of the viscosity parameters on the soil radial displacement and the pore water pressure are remarkable; while the viscosity parameters have a little influence on the circumferential stress of soil.