In this paper, we thoroughly analyze the linearized version of a poromechanics model developed to simulate biological tissues perfusion. This is a fully dynamical model in which the fluid and solid equations are strongly coupled through the interstitial pressure. As such, it generalizes Darcy, Brinkman and Biot equations of poroelasticity. The mathematical and numerical analysis of this model was first performed for a compressible porous material. Here, we focus on the nearly incompressible case with a semigroup approach that also enables to prove the existence of weak solutions. We show the existence and uniqueness of strong and weak solutions in the incompressible limit, for which a non-standard divergence constraint arises. Due to the special form of the coupling, the underlying problem is not coercive. Nevertheless, by using the notion of T-coercivity, we obtain stability estimates and well-posedness results. Our study also provides guidelines to propose a stable and robust approximation of the problem with mixed finite elements. In particular, we recover an inf-sup condition independent of the phase field. Finally, we investigate numerically the elliptic regularity of the associated steady-state problem and illustrate the sensitivity of the solution with respect to the different model parameters.