The study and the determination of polyhedral regions of local stability for linear systems subject to control saturation is addressed. The analysis of the nonlinear behavior of the closed-loop saturated system is made by dividing the state space in regions of saturation. Inside each of these regions, the system evolution can be represented by a linear system with an additive disturbance. From this representation, a necessary and sufficient condition relative to the contractivity of a given convex compact polyhedral set is stated. Consequently, the polyhedral set can be associated with a Lyapunov function and the local asymptotic stability of the saturated closed-loop system inside the set is guaranteed. Furthermore, it is shown how, in some particular cases, the compactness condition can be relaxed in order to ensure the asymptotic stability in unbounded polyhedra. Finally, an application of the contractivity conditions is presented in order to determine local asymptotic stability regions for the closed-loop saturated system.