We investigate 𝜙 meson photoproduction on the nucleon and the He 4 targets within a dynamical model approach based on a Hamiltonian which describes the production mechanisms by the Pomeron-exchange, meson-exchanges, 𝜙 radiations, and nucleon resonance excitations mechanisms. The final 𝜙𝑁 interactions are included being 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 𝛾 𝑝 → 𝜙𝑝 from the CLAS Collaboration. The resulting Hamiltonian is then used to predict the coherent 𝜙-meson production on the He 4 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 He 4 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.