Moving-horizon (MH) state estimation is addressed for nonlinear discrete-time systems affected by bounded noises acting on system and measurement equations by minimizing a sliding-window least-squares cost function. Such a problem is solved by searching for suboptimal solutions for which a certain error is allowed in the minimization of the cost function. Nonlinear parameterized approximating functions such as feedforward neural networks are employed for the purpose of design. Thanks to the offline optimization of the parameters, the resulting MH estimation scheme requires a reduced online computational effort. Simulation results are presented to show the effectiveness of the proposed approach in comparison with other estimation techniques.