The aim of this paper is to give some approximation results for a class of nonlinear filtering problems with delay in the observation. First, we point out some general results on the approximation problem for the filter in nonlinear filtering. In particular, we give a general procedure for obtaining some upper bounds for the different approximations we consider. This procedure is then applied in the case of nonlinear filtering problems with delay (X, Y), which can be represented by means of a Markov system (X, (Y) over cap ), in the sense that Y-t = (Y) over capa(t). Finally, these upper bounds are computed explicitly in the particular case of Markov jump process with counting observations.