This paper studies the phase transitions of Cu and Ni alloys as they cool from melting temperature to room temperature under high-pressure conditions. The interatomic forces acting between the atoms are modeled by the tight-binding potential. Control over the environmental pressure and the cooling temperature is maintained by a canonical ensemble (N, P, T) system. The numerical results confirm that the metal phase transition is influenced significantly by the pressure conditions, even in the case of pure Cu and Ni metals. Three specific transition pathways are identified for the Cu and Ni alloys as they cool from melting temperature to room temperature, namely a transition at the melting temperature to a crystalline structure, a transition at the glass transition temperature to a glass (amorphous) structure, and finally solidification at the melting temperature followed by a subsequent transition at the glass transition temperature. The results reveal that glass transition generally occurs at lower pressures in alloys with higher Cu compositions, while glass transition following prior solidification tends to takes place at higher pressures in alloys with higher Ni compositions.