In the limit of an approximate $\mu-\tau$ symmetry in the neutrino mass matrix, we explore deviations to the Tri-Bi-Maximal mixing pattern in the neutrino sector. We consider two different ansatzes for the corrected pattern to predict the current values of neutrino mixing parameters. We show that it is possible to constrain the Majorana $CP$ phases by studying their correlation to the mixing parameters and we study their effects on neutrinoless double beta decay observables. These predictions are sharp for the quasi-degenerate ordering and can be tested in upcoming experiments.