The process γγ→φ/ψ is measured using a data sample of 825fb-1 collected with the Belle detector. A narrow peak of 8.8-3.2+4.2 events, with a significance of 3.2 standard deviations including systematic uncertainty, is observed. The mass and natural width of the structure [named X(4350)] are measured to be [4350.6-5.1+4.6(stat)±0.7(syst)]MeV/ c2 and [13-9+18(stat)±4(syst)]MeV, respectively. The product of its two-photon decay width and branching fraction to φJ/ψ is [6.7-2.4+3.2(stat)±1.1(syst)]eV for JP=0+, or [1.5-0.6+0.7(stat)±0.3(syst)]eV for JP=2+. No signal for the Y(4140)→φJ/ψ structure reported by the CDF Collaboration in B→K+φJ/ψ decays is observed, and limits of Γγγ(Y(4140))B(Y(4140)→φJ/ψ)<41eV for JP=0 + or <6.0eV for JP=2+ are determined at the 90% C.L. This disfavors the scenario in which the Y(4140) is a Ds*+Ds*- molecule.