A coupled-channel model with \(\pi N\), \(\rho N\) and \(J/\Psi N\) channels is developed to predict the \(\pi + N \rightarrow J/\Psi + N \) cross sections. The \(J/\Psi\)-\(N\) interaction is parameterized in a form related to what has been predicted by the effective field theory approach and Lattice QCD. The other interactions within the model are constrained by the decay width of \(J/\Psi \rightarrow \rho + \pi\) and the total cross section data of \(\pi N\) reactions. The calculated meson-baryon amplitudes are then used to predict the cross sections of the \(J/\Psi\) production on the deuteron target by including the contributions from the impulse term and the one-loop calculations of the final \(NN\) and \(J/\Psi N\) re-scattering effects. Predictions of the dependence of the cross sections of \(\pi^- + p \rightarrow J/\Psi+ n\), \(\gamma + d \rightarrow J/\Psi + n + p\), and \(\pi^+ d \rightarrow J/\Psi +p +p\) on the \(J/\Psi\)-\(N\) potentials are presented for experimental determinations of the \(J/\Psi\)-\(N\) interaction. Within the vector meson dominance model, we have also applied the constructed coupled-channel model to predict the \(\gamma + p\rightarrow J/\Psi + p\) cross sections near the \(J/\Psi\) production threshold.