  Using a sample of $(10087\pm 44)\times10^{6}$ $J/\psi$ events collected by
the BESIII detector at the BEPCII collider, we search for the decay $X(1870)\to
K^+ K^-\eta$ via the $J/\psi\to \omega K^+ K^- \eta$ process for the first
time. No significant $X(1870)$ signal is observed. The upper limit on the
branching fraction of the decay $ J/\psi\to \omega X(1870) \to\omega K^+ K^-
\eta$ is determined to be $9.55\times 10^{-7}$ at the $90\%$ confidence level.
In addition, the branching faction $B(J/\psi\to\omega K^+ K^- \eta)$ is
measured to be $(3.33\pm0.02(\rm{stat.})\pm 0.12(\rm{syst.}))\times 10^{-4}$.
