Using data samples with an integrated luminosity of 6.4 fb(-1) collected by the BESIII detector operating at the BEPCII storage ring, the process of e+e- gamma phi J/psi is studied. The processes of e(+)e(-) -> phi chi c1,c2, chi c1,c2 gamma J/psi are observed with a significance of more than 10 sigma. The ,Vs-dependent cross section of e(+)e(-) -> phi chi(c1,c2) is measured between 4.600 and 4.951 GeV, and evidence of a resonance structure is found for the first time in the phi chi(c2) process. We also search for the processes of e+e- -> gamma X(4140), gamma X(4274) and gamma X(4500) via the gamma phi J/psi final state, but no obvious structures are found. The upper limits on the production cross section times the branching fraction for these processes at the 90% confidence level are reported.