We study the seesaw realization of seven textures of the neutrino mass matrix with two zeros, which were presented by Frampton, Glashow and Marfatia. It is found that once the Dirac neutrino mass matrix is fixed, only texture A2 out of the seven textures is realized by the seesaw with three different heavy Majorana right-handed neutrino (RHN) mass matrices. As a consequence of the seesaw mechanism, the out of equilibrium decay of the lightest RHN generates a lepton number asymmetry. By the leptogenesis process, this lepton asymmetry is converted to the Baryon Asymmetry of the Universe (BAU). By a proper choice of the parameter space of the model which satisfied the low energy experimental data, the BAU is successfully explained by both flavor dependent and flavor independent leptogenesis of the model. It is also shown that the predictions of Dirac charge parity (CP) violation phase, ?CP, for some fixed parameters of high energy physics can be constrained by the current observation of baryon asymmetry.