  The ambiguity in the choice of a renormalization scheme and scale for the
top-quark mass leads to an additional source of theoretical uncertainty in the
calculation of the Higgs boson production cross section via gluon fusion. These
uncertainties were found to be dominant in the case of off-shell Higgs
production at next-to-leading order in QCD for large values of the Higgs
virtuality $m_H^*$. In this work, we study the uncertainties related to the
top-quark mass definition up to next-to-next-to-leading order (NNLO) in QCD. We
include the full top-quark mass dependence up to three loops in the virtual
corrections, and evaluate the real contributions in the soft limit, therefore
obtaining the so-called soft-virtual (SV) approximation. We construct NNLO-SV
predictions for off-shell Higgs boson production renormalizing the top-quark
mass within both the on-shell (OS) and the MSbar schemes, and study in detail
the differences between them. While the differences between the two schemes are
sizeable, we find that the predictions are always compatible within scale
uncertainties. We also observe that the difference between renormalization
schemes is largely reduced when increasing the order of the perturbative
expansion. We analyze the quality of the convergence of the perturbative series
in both schemes, and find that at large invariant masses the MSbar results
present much larger corrections than their OS counterparts. We also comment on
the more complicated case of Higgs boson pair production.
