The effect of the linear polarization of gluons on the transverse momentum distribution in Higgs production is studied within the framework of TMD factorization. For this purpose we consider the TMD evolution for general colorless scalar boson production, from the lower mass C-even scalar quarkonium states chi(c0) and chi(b0) to the Higgs mass scale. In the absence of an intrinsic nonperturbative linearly polarized gluon distribution the results correspond to the CSS formalism, indicating a rather rapid decrease with increasing energy scale. At the Higgs mass scale the contribution from linearly polarized gluons is in this case found to be on the percent level, somewhat larger than an earlier finding in the literature. At the lower mass scale of quarkonium states chi(c0) and chi(b0) we find contributions at the 15-70% level, albeit with considerable uncertainty. In the presence of an intrinsic linear gluon polarization, percent level effects are also found at the Higgs mass scale, but with a considerably slower evolution. Although these results were obtained using a model for the TMDs that are approximately Gaussian at small transverse momenta and have the correct perturbative power law fall-off at large transverse momenta, it illustrates well the differences that can exist between results obtained from a TMD formalism as compared to a CSS formalism. The behavior of the TMDs at small PT can affect the results for all transverse momenta of the produced boson, even for a particle as heavy as the Higgs. The TMD evolution from chi(c0) to chi(b0) may be used to constrain the nonperturbative contributions and improve on the prediction of the effect at the Higgs mass scale. (C) 2014 The Authors. Published by Elsevier B.V.