The production of the hidden-charm pentaquarks P c via pion-induced reaction on a proton target is investigated within an effective Lagrangian approach. Three experimentally observed states, P c (4312), P c (4440), and P c (4457), are considered in the calculation, and the Reggeized t-channel meson exchange is considered as main background for the reaction π − p → J/ψn. The numerical results show that the experimental data of the total cross section of the reaction π − p → J/ψn at W ≃ 5 GeV can be well explained by contribution of the Reggeized t channel with reasonable cutoff. If the branching ratios Br[P c → J/ψN] ≃ 3% and Br[P c → πN] ≃ 0.05% are taken, the average value of the cross section from the P c (4312) contribution is about 1.2 nb/100 MeV, which is consistent with existing rude data at near-threshold energies. The results indicate that the branching ratios of the P c states to the J/ψN and πN should be small. The shape of differential cross sections shows that the Reggeized t-channel provides a sharp increase at extreme forward angles, while the differential cross sections from the P c states contributions are relatively flat. High-precision experimental measurements on the reaction π − p → J/ψn at near-threshold energies are suggested to confirm the LHCb hidden-charm pentaquarks as genuine states, and such experiments are also helpful to understand the origin of these resonance structures.