LHC provides a chance to study the physical processes of γγ collision and γ p collision mode by adding a forward detector. In this paper, we investigate J/ψ-pair photoproduction in NRQCD framework at the LHC with γγ collision mode. We calculate the contributions from ${~}^{3}S_{1}^{1}$ , ${~}^{3}S_{1}^{(8)}$ , ${{~}^{1}}S_{0}^{(8)}$ , ${~}^{3}P_{J}^{(8)}$ Fock states, and find that contribution from the colour octet ${{~}^{1}}{S_{0}^{8}}$ ${~}^{3}P_{J}^{(8)}$ and ${~}^{3} P_{J}^{(8)}{~}^{3}P_{J}^{(8)}$ channels dominates this process with forward detector acceptance 0.0015 < ξ < 0.5. We present the differential cross section distributions of transverse momentum $p_{t}^{J/\psi }$ and rapidity yJ/ψ with forward detector acceptances 0.0015 < ξ < 0.5, analyse the signal for J/ψ decaying to a opposite-sign (OS) muon pair. The results show that this process has the potential to be detected at the 13 TeV LHC with forward detector acceptance 0.0015 < ξ < 0.5.