The Standard Model of Particle Physics (SM) is agnostic in the source and nature of the neutrino masses. Indeed, there is some pedantic debate over whether neutrino mass is really part of the SM although from a practical perspective, most people would say that it is when they are talking about SM physics predictions.
If neutrinos did couple to the Higg field (which is the source of the rest mass of all of the other massive fundamental particles in the SM), they would have such a weak coupling that we would probably never observe Higgs boson decays to neutrinos even if they did, like other SM fundamental particles, have a coupling to the Higg field proportional to their rest mass.
I'll illustrate this conclusion with the following back of napkin class calculations.
The branching faction of the charged leptons is for tau-lepton pairs, 6.27% (observed), for muon pairs, 0.021 8% (observed), and for electron-positron pairs, 0.000 000 5% (not yet observed).
Roughly speaking branching fractions ratios are on the same order of magnitude of the square of mass ratios.
Electrons are 511,000 eV v. something on the order of 0.050 eV for the largest neutrino mass eigenstate, a ratio of 10.
This implies that if neutrinos got their mass via the same SM Higgs mechanism that applies to other SM fermions (whether or not that makes sense for other reasons), that a branching fraction from Higgs boson decays for neutrinos on the order of 10 smaller than that of the roughly 5*10 for electrons.
This would imply a Higgs boson branching fraction on the order of not more than 10 for any kind of neutrino, when muon pairs with a branching fraction of about 2*10 are at the current experimental detection threshold.
In addition, a smaller proportion of neutrinos passing through a detector are actually seen than a proportion of charged leptons passing through a detector, since neutrinos interact more weakly with other matter, so a Higgs boson decay into neutrinos would not be possible at the same branching fraction that a charge lepton Higgs boson decay can be detected.
Realistically, detector precision would have to improve by a factor of about 10 or more over current technology to directly observe neutrino decays from Higgs bosons in a statistically significant way.
I don't expect that to happen anytime during the lives of anyone who ever encounters me alive.