  We present estimates of transverse single spin asymmetry (TSSA) in
$p+p^\uparrow \rightarrow J/\psi+X$ within the colour evaporation model of
charmonium production in a generalized parton model (GPM) framework, using the
recently obtained best fit parameters for the gluon Sivers function (GSF)
extracted from PHENIX data on TSSA in $p+p^\uparrow \to \pi^0+X$ at
midrapidity. We calculate asymmetry at $\sqrt{s} = 200$ GeV, and compare the
results with PHENIX data on TSSA in the process $p + p^\uparrow \to J/\psi+X$.
We also present estimates for asymmetry at $\sqrt{s} = 115 $ GeV corresponding
to the proposed fixed target experiment AFTER@LHC and at $\sqrt{s} = 500$ GeV
corresponding to the higher RHIC energy. Finally, we investigate the effect of
the transverse momentum dependent (TMD) evolution of the densities involved, on
the asymmetry.
