This note addresses tracking control of nonlinear discrete-time monotone systems subject to input and state constraints. Forcing saturation on a previously designed controller may, in general, lead to destabilization or, at least, result in constraint violation and performance losses. Hereby, it is shown that for a certain class of nonlinear monotone systems it is possible to design a static nonlinear output feedback which, saturated among suitable state-dependent bounds, is able to guarantee constraint satisfaction and asymptotic tracking of piecewise constant references, with a moderate online computational burden. Simulation experiments concerning the synthesis of a protein demonstrate the effectiveness of the proposed control strategy.