  In this article, we investigate the $\cos2\phi_t$ azimuthal asymmetry in $e
~p\rightarrow e ~J/\psi ~Jet~ X$, where the $J/\psi$-jet pair is almost
back-to-back in the transverse plane, within the framework of the generalized
parton model(GPM). We use non-relativistic QCD(NRQCD) to calculate the $J/\psi$
production amplitude and incorporate both color singlet(CS) and color octet(CO)
contributions to the asymmetry. We estimate the asymmetry using different
parameterizations of the gluon TMDs in the kinematics that can be accessed at
the future electron-ion collider (EIC) and also investigate the impact of
transverse momentum dependent (TMD) evolution on the asymmetry. We present the
contributions coming from different states to the asymmetry in NRQCD.
