  We calculate the transverse single-spin asymmetry (SSA) for $J/\psi$
production in proton-proton collisions, within non-relativistic QCD, employing
the transverse momentum dependent, generalized parton model, which includes
both spin and intrinsic motion effects. In particular, we study the role of the
color octet mechanism in accessing the gluon Sivers function. In doing that, we
also show, within this approach, how the singularities coming from color octet
terms in the low-$P_T$ region can be handled, leading to finite cross sections.
Predictions for both unpolarized cross sections and SSAs are given and compared
against PHENIX data. Estimates for LHCb in the fixed target mode are also
presented.
