Summary: We perform a systematic investigation of the texture structures of Majorana neutrino mass matrix \(M_\nu\) having two texture zeros and an equality between two nonzero matrix elements, in the light of recent neutrino oscillation data. Among forty-two possible textures, it is found that only eight textures are compatible with the current experimental data at \(3\sigma\) C.L. Out of these phenomenologically viable textures, six follow normal mass ordering while remaining two satisfy the inverted mass ordering of neutrino mass spectrum. In the numerical analysis, we carry out a scan over the possible space of all viable patterns. We present the implications of each allowed patterns for three mixing angles (solar, reactor and atmospheric), leptonic CP-violation, neutrino mass scale and the neutrinoless double beta decay indicating strong correlations between oscillation parameters. The symmetry realization of one of the viable textures is also presented.