We consider nonlinear discrete-time control system with linearizable right-hand side and state and control constraints. The problem is to estimate the finite-time controllability region of the system, i. e., the set of initial states for which there is an admissible control sequence such that it moves the system from the initial point to the predefined terminal region in a given number of steps. An algorithm to estimate the controllability region is proposed. It is based on the property of linearizable systems to map a small polyhedron into approximately a polyhedron. The algorithm estimates the controllability region with prescribed accuracy by a union of polyhedra.