The process gamma gamma -> phi J/psi is measured using a data sample of 825 fb(-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/c(2) and [13(-9)(+18)(stat) +/- 4(syst)] MeV, respectively. The product of its two-photon decay width and branching fraction to phi J/psi is 6.7(-2.4)(+3.2) (stat) +/- 1.1(syst)] eV for J(P) = 0(+), or [1.5(-0.6)(+0.7) (stat) +/- 0.3 (syst)]eV for J(P) = 2(+). No signal for the Y(4140) ->phi J/psi structure reported by the CDF Collaboration in B -> K+ phi J/psi decays is observed, and limits of Gamma(gamma gamma) (Y(4140))B(Y(4140) -> phi J/psi) < 41 eV for J(P) = 0(+) or <6.0 eV for J(P) = 2(+) are determined at the 90% C. L. This disfavors the scenario in which the Y(4140) is a D-s*D-+(s)*(-) molecule.