  We calculate azimuthal asymmetries in back-to-back production of $J/\psi$ and
a photon in electron-proton scattering process at the future electron-ion
collider (EIC) using TMD factorization framework. We consider the cases where
the proton is unpolarized or transversely polarized. For the formation of
$J/\psi$, non-relativistic QCD (NRQCD) is used. We find that the cross-section
gets contribution from only one color octet state, as a result, the azimuthal
modulations become independent of the long-distance matrix elements (LDMEs),
and thus can be used to probe, in particular the gluon TMDs which are dominant
in this kinematics. We show estimates of the upper bounds of different
azimuthal asymmetries using the positivity bounds on TMDs. In addition, we show
estimate of asymmetries using the Gaussian parametrization of the TMDs.
