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 $m_ u$ and charged lepton mass matrix $m_e$. If we restrict to the minimally parametrized non-singular `$m_e$ (i.e., with maximum number of zeros) it gives rise to only 6 possible textures of $m_e$. Non-zero determinant of $m_ u$ dictates six possible textures of the constituent matrices. We ask in this minimalistic approach, what are the 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 $mu_S$ to obtain all 7 viable two-zero textures of $m_ u$. 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 ($m_ u$). 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 $delta_{CP}$. Therefore, if $delta_{CP}=pi/2$ is established, our minimalistic scheme may still be viable provided we allow more number of parameters in `$m_e$.
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