A new model of compressible multiphase flow in a deforming porous medium under finite deformations is presented. The derivation of the model is based on the application of the theory of a Symmetric Hyperbolic Thermodynamically Compatible (SHTC) systems to a multiphase mixture of solid and fluids. The viscosity of the saturating fluid can be taken into account when describing the fluid by Maxwell's nonlinear relaxation model. The formulated governing PDEs are hyperbolic and satisfy the laws of irreversible thermodynamics - conservation of energy and growth of entropy. Due to the above properties, the formulated model is well suited for the straightforward application of modern high accuracy numerical methods applicable to the solution of hyperbolic systems, and ensures the reliability of solutions obtained numerically. On the basis of a nonlinear model, the governing equations for the propagation of small-amplitude waves are derived, which form linear system of hyperbolic equations written in terms of velocities, relative velocities, pressure and stress deviator. The latter makes it possible to use an efficient finite-difference scheme on staggered grids for their numerical solution. Some numerical test problems are presented showing the features of wave propagation in a porous medium saturated with two different liquids.