Possibilities of observing a signal of an η n bound state are investigated by considering photoproductions of the η and η mesons on a deuteron target with forward proton emission. For this purpose, we take the η n interaction from the linear σ model with a coupling to ηn, in which an s-wave η n bound state can be dynamically generated, and we fix the γp → ηp and η p scattering amplitudes so as to reproduce the experimental cross sections with forward proton emission. By using these γp → η ( ) p and η ( ) n → η ( ) n amplitudes, we calculate cross sections of the γ d → ηnp and η np reactions with forward proton emission in single and η ( ) -exchange double-scattering processes. As a result, we find that the signal of the η n bound state can be seen below the η n threshold in the ηn invariant mass spectrum of the γ d → ηnp reaction and is comparable with the contribution from the quasifree η production above the η n threshold. We also discuss the behavior of the signal of the η n bound state in several experimental conditions and model parameters.