Binary Mg-Zn and Al-Zn alloys have been investigated theoretically under static isotropic pressure. The stable phases of these binaries on both initially hexagonal-close-packed (hcp) and face-centered-cubic (fcc) lattices have been determined by utilizing an iterative approach that uses a configurational cluster expansion method, Monte Carlo search algorithm, and density functional theory (DFT) calculations. Based on 64-atom models, it is shown that the most stable phases of the Mg-Zn binary alloy under ambient condition are MgZn3, Mg19Zn45, MgZn, and Mg34Zn (30) for the hcp lattice, and MgZn3 and MgZn for the fcc lattice, whereas the Al-Zn binary is energetically unfavorable throughout the entire composition range for both the hcp and fcc lattice symmetries under all pressure conditions. By applying an isotropic pressure in the hcp lattice, Mg19Zn45 turns into an unstable phase at P approximate to 10 GPa, a new stable phase Mg3Zn appears at P greater than or similar to 20 GPa, and Mg34Zn30 becomes unstable for P greater than or similar to 30 GPa. For the fcc lattice, the Mg3Zn phase weakly touches the convex hull at P greater than or similar to 20 GPa while the other stable phases remain intact up to approximate to 120 GPa. Furthermore, making use of the obtained DFT results, the bulk modulus has been computed for several compositions up to pressure values on the order of approximate to 120 GPa. The findings suggest that one can switch between Mg-rich and Zn-rich early-stage clusters simply by applying external pressure. Zn-rich alloys and precipitates are more favorable in terms of stiffness and stability against external deformation.