Using (223.7±1.4)x10^6 J/psi events accumulated with the BESIII detector, we study eta_c decays to phi phi and phi omega final states. The branching fraction of eta_c --> phi phi is measured to be Br(eta_c --> phi phi)=(2.5±0.3-0.7+0.3±0.6)x10^-3, where the first uncertainty is statistical, the second is systematic, and the third is from the uncertainty of Br(J/psi --> gamma eta_c). No significant signal for the double Okubo-Zweig-Iizuka suppressed decay of eta_c --> omega phi is observed, and the upper limit on the branching fraction is determined to be Br(eta_c --> omega phi)<2.5x10^-4 at the 90% confidence level