Ti alloys exhibit exceptional performance owing to their low density and high specific strength. Consequently, they have been employed in a wide range of applications, including medical devices and aerospace components. In recent years, Ti alloys have attracted considerable attention as biomaterials for artificial bone implants due to their excellent biocompatibility. It is well established that the addition of alloying elements such as Nb, V or Mo to Ti can yield reduced elastic stiffness, particularly with regards to Young’s modulus. To understand the phase stability and mechanical properties of β-Ti alloys, first-principle calculations were performed on B2 Ti–Mo alloys using a supercell approach comprising 48 atoms. The lattice parameters of the cubic TiMo alloy are in good agreement with both theoretical predictions and previously reported experimental data. In addition, the calculated heats of formation indicate that the TiMo alloy is thermodynamically stable, as evidenced by their negative formation energies. Furthermore, the computed elastic constants demonstrate that the TiMo alloy is mechanically stable, since it satisfies the mechanical stability criteria for a cubic crystal system. The Young’s modulus agrees well with expected values, and both Pugh’s ratio and Poisson’s ratio have been evaluated to further characterize the mechanical response. The calculated phonon dispersion relations reveal that TiMo is vibrationally stable, as indicated by the absence of soft modes. Therefore, these elastic and vibrational characteristics suggest that the TiMo alloys are promising candidates for future biomedical applications.