  We propose two different seesaw models namely, type I and inverse seesaw to
realise light Dirac neutrinos within the framework of $A_4$ discrete flavour
symmetry. The additional fields and their transformations under the flavour
symmetries are chosen in such a way that naturally predicts the hierarchies of
different elements of the seesaw mass matrices in these two types of seesaw
mechanisms. For generic choices of flavon alignments, both the models predict
normal hierarchical light neutrino masses with the atmospheric mixing angle in
the lower octant. Apart from predicting interesting correlations between
different neutrino parameters as well as between neutrino and model parameters,
the model also predicts the leptonic Dirac CP phase to lie in a specific range
-$\pi/3$ to $\pi/3$. While the type I seesaw model predicts smaller values of
absolute neutrino mass, the inverse seesaw predictions for the absolute
neutrino masses can saturate the cosmological upper bound on sum of absolute
neutrino masses for certain choices of model parameters.
