A feedback linearizing control strategy for discrete-time nonlinear systems subject to input constraints is proposed. The control system comprises : (i) a feedback linearizing controller; (ii) a constraint mapping algorithm that transforms the actual input constraints into constraints on the feedback linearized system; and (iii) a linear model predictive controller that regulates the resulting constrained linear system. Closed-loop stability analysis is a challenging problem because the transformed constraints are state dependent. Sufficient conditions for asymptotic stability are presented for fully and partially feedback linearizable systems. As part of the analysis, a new stability result for unconstrained discrete-time nonlinear systems which parallels a well-known continuous-time result is derived.