  One of the main goals of future electron-ion colliders is to improve our
understanding of the structure of hadrons. In this letter, we study the
exclusive $\eta_c$ production by $\gamma^{*} \gamma$ interactions in $eA$
collisions and demonstrate that future experimental analysis of this process
can be used to improve the description of the $\eta_c$ transition form factor.
The rapidity, transverse momentum and photon virtuality distributions are
estimated considering the energy and target configurations expected to be
present at the EIC, EicC and LHeC and assuming different predictions for the
light-front wave function of the $\eta_c$ meson. Our results indicate that the
electron-ion colliders can be considered an alternative to providing
supplementary data to those obtained in $e^- e^+$ colliders.
