We investigate ϕ meson photoproduction on the nucleon and 4He targets within a dynamical model approach based on a Hamiltonian which describes the production mechanisms by the Pomeron exchange, meson exchange, ϕ radiation, and nucleon resonance excitation mechanisms. The final ϕN interactions are included and described by the gluon-exchange, direct ϕN couplings, and the box diagrams arising from the couplings with πN, ρN, KΛ, and KΣ channels. The parameters of the Hamiltonian are determined by the experimental data of γp→ϕp from the CLAS Collaboration. The resulting Hamiltonian is then used to predict the coherent ϕ-meson production on the 4He targets by using the distorted-wave impulse approximation. For the proton target, the final ϕN rescattering effects, as required by the unitarity condition, are found to be very weak, which supports the earlier calculations in the literature. For the 4He targets, the predicted differential cross sections are in good agreement with the data obtained by the LEPS Collaboration. The role of each mechanism in this reaction is discussed and predictions for a wide range of scattering angles are presented, which can be tested in future experiments.