  We discuss the impact on the study of gluon transverse momentum dependent
distributions (TMDs) of the associated production of a lepton pair and a
$\Upsilon$ (or a $J/\psi$) in unpolarised proton-proton collisions, $pp\to
{\cal Q} \, \ell \bar{\ell} X$, at LHC energies, where one can assume that such
final states are dominantly induced by gluon fusion. If the transverse momentum
of the quarkonium-dilepton system - namely, the transverse momentum imbalance
of the quarkonium state and the lepton pair - is small, the corresponding cross
sections can be calculated within the framework of TMD factorisation. Using the
helicity formalism, we show in detail how these cross sections are connected to
the moments of two independent TMDs: the distribution of unpolarised gluons,
$f_1^g$, and the distribution of linearly polarised gluons, $h_1^{\perp g}$. We
complete our exhaustive derivation of these general relations with a
phenomenological analysis of the feasibility of the TMD extraction, as well as
some outlooks.
