Using a sample of (10087 +/- 44) x 10(6) J/psi events collected by the BESIII detector at the BEPCII collider, we search for the decay X(1870) -> K+K-eta via the J/psi -> omega K+K-eta process for the first time. No significant X(1870) signal is observed. The upper limit on the branching fraction of the decay J/psi -> omega X(1870) -> omega K+K-eta is determined to be 9.55 x 10(-7) at the 90% confidence level. In addition, the branching faction B(J/psi -> omega K+K-eta) is measured to be (3.33 +/- 0.02(stat) +/- 0.12(syst) x 10(-4).