  We calculate the Sivers asymmetry in the photoproduction of almost
back-to-back $J/\psi$-jet pair in the process $ep^\uparrow \to
J/\psi+\mathrm{jet}+X$, which will be possible at the future planned
electron-ion collider (EIC). We use the framework of generalized parton model
(GPM), and NRQCD for calculating the $J/\psi$ production rate. We include
contributions from both color singlet and color octate states in the asymmetry.
We obtain sizable Sivers asymmetry that can be promising to determine the gluon
Sivers function. We also investigate the effect of TMD evolution on the
asymmetry.
