  Recently it has been put forward that linearly polarized gluons inside
unpolarized protons affect the transverse momentum distribution of final state
particles in hadronic collisions. They lead to a characteristic modulation of
the differential cross section in Higgs production and to azimuthal asymmetries
in, for instance, heavy quark pair production. Here we study the effect on
charmonium and bottomonium production in hadronic collisions, such as at LHCb
and at the proposed fixed target experiment AFTER at LHC. We focus mainly on
the scalar and pseudoscalar quarkonia, eta_c, chi_{c0}, eta_b, chi_{b0}, which
allow for an angular independent investigation. Within the framework of
transverse momentum dependent factorization in combination with the
nonrelativistic QCD based color-singlet quarkonium model, we show for small
transverse momentum (q_T^2 \ll 4 M_Q^2) that the scalar and pseudoscalar
quarkonium production cross sections are modified in different ways by linearly
polarized gluons, while their effects on the production of higher angular
momentum quarkonium states are strongly suppressed. Comparisons to chi_{c2},
chi_{b2} production can help to cancel out uncertainties. Together with the
analogous study in Higgs production at LHC, quarkonium production can moreover
be used to test the scale dependence of the linearly polarized gluon
distribution over a large energy range.
