  We study the $\eta^\prime g^* g^*$ vertex for arbitrary gluon virtualities in
the time-like and space-like regions, using the perturbative QCD hard
scattering approach and an input wave-function of the $\eta^\prime$-meson
consistent with the measured $\eta^\prime \gamma^* \gamma$ transition form
factor. The contribution of the gluonic content of the $\eta^\prime$-meson is
taken into account, enhancing the form factor over the entire virtuality
considered. However, data on the electromagnetic transition form factor of the
$\eta^\prime$-meson is not sufficient to quantify the gluonic enhancement. We
also study the effect of the transverse momenta of the partons in the
$\eta^\prime$-meson on the $\eta^\prime g^* g^*$ vertex, using the modified
hard scattering approach based on Sudakov formalism. Analytic expressions for
the $\eta^\prime g^* g^*$ vertex are presented in limiting kinematic regions
and parametrizations are given satisfying the QCD anomaly, for real gluons, and
perturbative QCD behavior for large gluon virtualities, in both the time-like
and space-like regions. Our results have implications for the inclusive decay
$B \to \eta^\prime X$ and exclusive decays, such as $B \to \eta^\prime
(K,K^*)$, and in hadronic production processes $N + N (\bar N) \to \eta^\prime
X$.
