  We study the transverse single-spin asymmetry (TSSA) in $p^{\uparrow}p \to
J/\psi X$ reaction, incorporating both transverse-momentum and spin effects. To
predict production cross section of prompt $J/\psi$ we use two different
approaches, the non-relativistic QCD (NRQCD) factorization approach and the
Improved Color Evaporation Model (ICEM), and show how the predicted results for
TSSAs depend on choice of hadronization model. For initial-state factorization
we consider two models: the standard Generalized Parton Model (GPM) and the
Colour Gauge-Invariant version of it (CGI-GPM). We demonstrate that PHENIX
collaboration data on TSSA in the process $p^{\uparrow}p \to J/\psi X$
constrain the gluon Sivers function of the proton and rule-out one of existing
parameterizations. Estimates for the TSSAs in $p^{\uparrow}p \to J/\psi X$
process for the conditions of the future SPD NICA experiment are presented for
the first time.
