We investigate the possibility that the process of p(0)-meson photoproduction on proton, gamma+p --> p+p(0), in the near threshold region E-gamma < 2 GeV, can be considered in the framework of model with pi-, sigma- and N-exchanges. This suggestion is based on a study of the t-dependence of differential cross section, dsigma(gammap --> pp(0))/dt, which has been measured by SAPHIR Collaboration. We find that the suggested model provides a good description of the experimental data with new values of pNN-coupling constants in the region of the time-like p(0)-meson momentum. Our results suggest that such model can be considered as a suitable nonresonant background mechanism for the future discussion of possible role of nucleon resonance contributions. Our predictions for p(0)-meson photoproduction on neutron target and for beam asymmetry on both proton and neutron targets are presented.