The pp¯ invariant mass spectra of the processes J/ψ→pp¯ω, J/ψ→pp¯ρ, and J/ψ→pp¯γ close to the pp¯ threshold are calculated by means of the NN¯ optical potential. The potential model for NN¯ interaction in the S01 state is proposed. The parameters of the model are obtained by fitting the cross section of NN¯ scattering together with the pp¯ invariant mass spectra of the J/ψ decays. Good agreement with the available experimental data is achieved. Using our potential and the Green's function approach we also describe the peak in the η′π+π− invariant mass spectrum in the decay J/ψ→γη′π+π− in the energy region near the NN¯ threshold.