A generalized inverse seesaw model, in which the 9 x 9 neutrino mass matrix has vanishing (1,1) and (1,3) submatrices, is proposed. This is similar to the universal two-zero texture which gives vanishing (1,1) and (1,3) elements of the 3 x 3 mass matrices in both the charged lepton and neutrino sectors. We consider the Z(6) x Z(6) group to realize such texture zeros in the framework of the generalized inverse seesaw model. We also analyze the universal two-zero texture in the general case and propose two Ansatze to reduce the number of free parameters. Taking account of the new result of theta(13) from the Daya Bay experiment, we constrain the parameter space of the universal two-zero texture in the general case and in the two Ansatze, respectively. We find that one of the Ansatze works well.