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We study the minimal type-III seesaw model in which we extend the SM byadding two $SU(2)_L$ triplet fermions with zero hypercharge to explain theorigin of the non-zero neutrino masses. We show that the naturalness conditionsand the limits from lepton flavor violating decays provide very stringentbounds on the model parameters along with the constraints from thestability/metastability of the electroweak vacuum. We perform a detailedanalysis of the model parameter space including all the constraints for bothnormal as well as inverted hierarchies of the light neutrino masses. We findthat most of the region that are allowed by lepton flavor violating decays andnaturalness fall in the stable/metastable region depending on the values of thestandard model parameters.

Constraining minimal Type-III Seesaw Model from naturalness, Lepton Flavor Violation and Electroweak Vacuum stability