The quasistatic Biot-Pressure system is a coupled system of parabolic and elliptic partial differential equations that describe the small deformations of and fluid flow through a fully saturated elastic and porous structure. It models the situation in which the inertia effects are negligible. This arises naturally in the classical Biot model of consolidation for a linearly elastic and porous solid which is saturated by a slightly compressible viscous fluid. Our objective is to extend the existence-uniqueness-regularity theory for such systems to include problems with constraints on the displacement. Such contact problems are highly nonlinear and ubiquitous in the applications.