  The scale of neutrino masses and the Planck scale are separated by more than
twenty-seven order of magnitudes. However, they can be linked by imposing the
stability of the electroweak (EW) vacuum. The crucial ingredient is provided by
the generation of neutrino masses via a seesaw mechanism triggered by Yukawa
interactions between the standard model (SM) Higgs and lepton doublets and
additional heavy right-handed neutrinos. These neutrinos participate to the
renormalization group (RG) running of the dimensionless SM couplings, affecting
their high-energy behavior. The Higgs quartic coupling is dragged towards
negative values, thus altering the stability of the EW vacuum. In the usual
type-I seesaw model, this effect is too small to be a threat since, in order to
comply with low-energy neutrino data, one is forced to consider either too
small Yukawa couplings or too heavy right-handed neutrinos. In this paper we
explore this general idea in the context of low-scale seesaw models. These
models are characterized by sizable Yukawa couplings and right-handed neutrinos
with mass of the order of the EW scale, thus maximizing their impact on the RG
flow. As a general result, we find that Yukawa couplings such that ${\rm
Tr}(Y_{\nu}^{\dag}Y_{\nu}) \gtrsim 0.4$ are excluded. We discuss the impact of
this bound on several observables, with a special focus on the lepton flavor
violating process $\mu\to e\gamma$ and the neutrino-less double beta decay.
