We report the results of a search for the X(1835) state in the process e(+)e(-) -> 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(Bom) (e(+)e(-) -> J/psi X(1835)).B(X(1835) ->>= 3 charged tracks) < 1.3 fb at 90% confidence level.