  We show how transverse momentum dependent gluon distributions could be probed
at a future Electron-Ion Collider through the analysis of transverse momentum
spectra and azimuthal asymmetries in the inclusive electroproduction of
$J/\psi$ and $\Upsilon$ mesons. The maximum values of these asymmetries,
obtained in a model-independent way by imposing the positivity bounds of the
polarized gluon distributions, suggest the feasibility of the proposed
measurements.
