  We constrain the Higgs-portal model employing the vector-boson fusion channel
at the LHC. In particular, we include the phenomenologically interesting
parameter region near the Higgs resonance, where the Higgs-boson mass is close
to the threshold for dark-matter production and a running-width prescription
has to be employed for the Higgs-boson propagator. Limits for the Higgs-portal
coupling as a function of the dark-matter mass are derived from the CMS search
for invisible Higgs-boson decays in vector-boson fusion at 13 TeV. Furthermore,
we perform projections for the 14 TeV HL-LHC and the 27 TeV HE-LHC taking into
account a realistic estimate of the systematic uncertainties. The respective
upper limits on the invisible branching ratio of the Higgs boson reach a level
of 2 % and constrain perturbative Higgs-portal couplings up to dark-matter
masses of about 110 GeV.
