We present a search for fJ(2220) production in radiative J/ψ→γfJ(2220) decays using 460fb-1 of data collected with the BABAR detector at the SLAC PEP-II e+e- collider. The fJ(2220) is searched for in the decays to K+K- and KS0KS0. No evidence of this resonance is observed, and 90% confidence level upper limits on the product of the branching fractions for J/ψ→γfJ(2220) and fJ(2220)→K+K-(KS0KS0) as a function of spin and helicity are set at the level of 10-5, below the central values reported by the Mark III experiment.