We have calculated the values of the critical packing fractions for the mixtures of symmetric nonadditive hard spheres at high densities for small values of the nonadditivity parameter. Calculations have been performed for solid solid and fluid fluid demixing transitions. A cluster algorithm for Monte Carlo simulations in a semigrand ensemble was used, and the waste recycling method was applied to improve the accuracy of the calculations. The finite size scaling analysis was employed to compute the critical packing fractions for infinite systems with high accuracy.