We report the results of a search for the $X(1835)$ state in the process $e^+e^-\to J/\psi+X(1835)$ using a data sample of 672 fb$^{-1}$ collected with the Belle detector at and near the $\Upsilon(4S)$ resonance at the KEKB asymmetric-energy $e^+e^-$ collider. No significant evidence is found for this process, and an upper limit is set on its cross section times the branching fraction: $\sigma_{\rm Born}(e^+e^- \to J/\psi X(1835)) \cdot$ {${\cal B}(X(1835)\to \ge 3$ charged tracks)} $< 1.3 \ {\rm fb}$ at 90% confidence level.
