We study the implications of a large value of the effective Majorana neutrino mass for a class of two-texture zero neutrino mass matrices in the flavour basis. We find that these textures predict near-maximal atmospheric neutrino mixing angle in the limit of large effective Majorana neutrino mass. We present the symmetry realization of these textures using the discrete cyclic group Z3. It is found that the texture zeros realized in this work remain stable under the renormalization group running of the effective neutrino mass matrix at one-loop level.
