In this paper, we study the thermodynamical and mathematical consistencies for a non-singular early-time viscous cosmological model known as soft-Big Bang, which was previously found in [N. Cruz, E. González and J. Jovel, Phys. Rev. D 105, 024047 (2022)]. This model represents a flat homogeneous and isotropic universe filled with a dissipative radiation fluid and a cosmological constant Γ, which is small but not negligible, in the framework of Eckart's theory. In particular, we discuss the capability of the solution in the fulfillment of the three following conditions: (i) the near equilibrium condition, which is assumed in Eckart's theory of non-perfect fluids, (ii) the mathematical stability of the solution under small perturbations, and (iii) the positiveness of the entropy production. We have found that this viscous model can describe the radiation domination era of the γCDM model and, at the same time, fulfill the three conditions mentioned by the fulfillment of a single constraint on the bulk viscous coefficient ζ0, finding also that this non-singular model has a positive energy density in the infinity past which is infinity hotter with a constant entropy.