We aim to explore the production rate of the pseudoscalar glueball in $J/psi$ radiative decay by lattice QCD in quenched approximation. The calculation is performed on three anisotropic lattices with the spatial lattice spacing ranging from 0.222(2) fm to 0.110(1) fm. As a calibration of some systematical uncertainties, we first extract the $M1$ form factor $hat{V}(0)$ of the process $J/psitogammaeta_{c}$ and get the result $hat{V}(0)=1.933(41)$ in the continuum limit, which gives the partial width $Gamma(J/psitogammaeta_{c})=2.47(11)$ keV. These results are in agreement with that of previous lattice studies. As for the pseudoscalar glueball $G_{0^{-+}}$, its mass is derived to be $2.395(14)$ GeV, and the form factor $hat{V}(0)$ of the process $J/psitogamma G_{0^{-+}}$ is determined to be $hat{V}(0)=0.0246(43)$ after continuum extrapolation. Finally, the production rate of the pseudoscalar glueball is predicted to be $2.31(90)times10^{-4}$, which is much smaller than that of conventional light $qbar{q}$ $eta$ states. After the subtraction of the phase space factor, the couplings of $J/psi Xgamma$ are similar where $X$ stands for $eta$ states and the pseudoscalar glueball. Possibly, the $U_{A}(1)$ anomaly plays an important role for the large couplings of gluons to the flavor singlet $eta$ states in $J/psi$ radiative decays.
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