The presence of unmeasured or uncontrolled inputs to dynamical systems can lead to instability or degraded performance. It becomes imperative, therefore, to estimate system states in conjunction with these exogenous disturbance inputs in order to design mitigative robust control strategies. In this paper, a systematic unknown input observer design methodology is proposed for discrete-time nonlinear systems with additive bounded disturbance inputs. Sufficient conditions in the form of linear matrix inequalities are proposed for constructing the observer gains, and providing levels of guaranteed state and unknown input estimation performance. The effectiveness of the proposed method is demonstrated on a numerical example.