The production of charmonium associated with Higgs boson via gluon fusion has been investigated in Ref.[Phys.Rev.D66,114002(2002)], in which they considered the contribution of final Higgs boson radiation off the charm quark at tree level and found that this process is to be far too rare to be observable in any of the considered experiments. In this paper, the production of prompt $J/psi$ associated with Higgs boson via gluon fusion at the 14 TeV LHC within the factorization formalism of NRQCD is revisited. After considering the contribution from the final Higgs boson radiation off the top quark in the loop, which is {more than} three orders of magnitudes over the charm quark at tree level, the production of prompt $J/psi$ associated with Higgs boson has great potential to be detected. The prompt $J/psi$ production includes the direct production and indirect production via radiative or hadronic decays of high excited charmonium states. For the direct $J/psi + H$ production via gluon fusion loop-induced, the ${}^{3}S^{(8)}_1$ Fock state gives dominant contribution to the cross section, which is about 95% to the total direct production. The indirect contribution via loop-induced is appreciable, since the summation of which from $psi(2S) + H$, $chi_{c1} + H$ and $chi_{c2} + H$ is about $34%$ to the total cross section of prompt $J/psi + H$. While the indirect contribution from $chi_{c0} + H$ is tiny, which can be neglected. With the great potential to be detected, prompt $J/psi$ production in associated with Higgs boson can help us to further understand the mechanism of colour-octet, as well as can be useful to further investigate the coupling of the Higgs boson and fermion.
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