  In this paper we consider Higgs plus jet production as a process that is
sensitive to the linear polarization of gluons inside the unpolarized protons
of the LHC. The leading order expressions for the transverse momentum
distribution of the Higgs plus jet pair are provided in terms of transverse
momentum dependent quark and gluon distributions. This includes both angular
independent and azimuthal angular dependent contributions, presented directly
in the laboratory frame. Lacking experimental constraints on the linearly
polarized gluon distribution, we study its effects on Higgs plus jet production
using two different models to illustrate the generic features and maximal
effects. It is found that the $\cos 2 \phi$ distribution may be the most
promising observable, as it is driven by only one initial linearly polarized
gluon. The potential advantages of the Higgs plus jet process compared to other
processes sensitive to the linear polarization of gluons are discussed.
