The cross sections of the processes e + e − → K + K − J/ψ and K 0 S K 0 S J/ψ are measured via initial state radiation at center-of-mass energies between the threshold and 6.0 GeV using a data sample of 980 fb −1 collected with the Belle detector on or near the Υ(nS) resonances, where n =1, 2, ..., 5. The cross sections for e + e − → K + K − J/ψ are at a few pb level and the average cross section for e + e − → K 0 S K 0 S J/ψ is 1.8 ± 0.6(stat.) ± 0.3(syst.) pb between 4.4 and 5.2 GeV. All of them are consistent with previously published results with improved precision. A search for resonant structures and associated intermediate states in the cross section of the process e + e − → K + K − J/ψ is performed.