  The naturalness of electroweak scale in the models of type-I seesaw mechanism
with ${\cal O}(1)$ Yukawa couplings requires TeV scale masses for the fermion
singlets. In this case, the tiny neutrino masses have to arise from the
cancellations within the seesaw formula which are arranged by fine-tuned
correlations between the Yukawa couplings and the masses of fermion singlets.
We motivate such correlations through the framework of discrete symmetries. In
the case of three Majorana fermion singlets, it is shown that the exact
cancellation arranged by the discrete symmetries in seesaw formula necessarily
leads to two mass degenerate fermion singlets. The remaining fermion singlet
decouples completely from the standard model. We provide two candidate models
based on the groups $A_4$ and $\Sigma(81)$ and discuss the generic
perturbations to this approach which can lead to the viable neutrino masses.
