The phenomenon of the near p (p) over bar -threshold enhancement observed in the J/psi -> gamma p (p) over bar decay is studied by using the enhancement factor method with a simpler one-pion-exchange potential between p and (p) over bar. The Jost function caused by the mentioned potential is perturbatively calculated in the zero-th order approximation, and the corresponding enhancement. factor is obtained. It is found that such a final state interaction offers an important contribution to the decay width near the p (p) over bar -threshold, although it is not large enough. To explain the decay data, a phenomenological factor G(p) with the form of 285500/(m(pi)(2) + p(2)) should be introduced. A further calculation including the p-dependent bare T-matrix, a more realistic N (N) over bar potential and the contribution from the higher-order wave functions would provide a better understanding of the decay data and even the existence of the baryonium p (p) over bar. The near p (p) over bar -threshold behavior of the decay width in the J/psi -> pi(0) p (p) over bar process is also discussed.