In this paper we consider Higgs boson 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 boson 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 boson 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 boson plus jet process compared to other processes sensitive to the linear polarization of gluons are discussed.