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We investigate Linear and Inverse seesaw mechanisms with maximal zero textures of the constituent matrices subjected to the assumption of non-zero eigenvalues for the neutrino mass matrix mν and charged lepton mass matrix me. If we restrict to the minimally parametrized non-singular ‘me’ (i.e., with maximum number of zeros) it gives rise to only 6 possible textures of me. Non-zero determinant of mν dictates six possible textures of the constituent matrices. We ask in this minimalistic approach, what phenomenologically allowed maximum zero textures are possible. It turns out that Inverse seesaw leads to 7 allowed two-zero textures while the Linear seesaw leads to only one. In Inverse seesaw, we show that 2 is the maximum number of independent zeros that can be inserted into μS to obtain all 7 viable two-zero textures of mν. On the other hand, in Linear seesaw mechanism, the minimal scheme allows maximum 5 zeros to be accommodated in ‘m’ so as to obtain viable effective neutrino mass matrices (mν). Interestingly, we find that our minimalistic approach in Inverse seesaw leads to a realization of all the phenomenologically allowed two-zero textures whereas in Linear seesaw only one such texture is viable. Next, our numerical analysis shows that none of the two-zero textures give rise to enough CP violation or significant δCP. Therefore, if δCP=π/2 is established, our minimalistic scheme may still be viable provided we allow larger number of parameters in ‘me’.