  We show that the production of a Higgs boson in association with a jet in
proton-proton collisions is sensitive to the linear polarization of gluons
inside unpolarized protons. We present the analytical expressions, at leading
order in perturbative QCD, for various transverse momentum dependent
observables, which can be measured at the LHC. Since there are no experimental
constraints on the linearly polarized gluon distribution, its effects are
studied by adopting two different models. In particular, we find that the
$\cos2\phi$ azimuthal asymmetry is a very promising observable because it could
give access to the sign of this new distribution.
