  We study the minimal type-III seesaw model in which we extend the SM by
adding two $SU(2)_L$ triplet fermions with zero hypercharge to explain the
origin of the non-zero neutrino masses. We show that the naturalness conditions
and the limits from lepton flavor violating decays provide very stringent
bounds on the model parameters along with the constraints from the
stability/metastability of the electroweak vacuum. We perform a detailed
analysis of the model parameter space including all the constraints for both
normal as well as inverted hierarchies of the light neutrino masses. We find
that most of the region that are allowed by lepton flavor violating decays and
naturalness fall in the stable/metastable region depending on the values of the
standard model parameters.
