We have investigated the temperature and pressure variation of the M₃ phonon mode of ReO₃ by inelastic neutron scattering on a single crystal. The energy of the M₃ phonon mode decreases with decreasing temperature contrary to the normal hardening behavior. The energy of this mode also decreases with increasing pressure and tends to zero at about 3.5 kbar at T=150 K. The M₃ mode softening drives the pressure-induced phase transition in ReO₃. The temperature-induced softening of the M₃ mode on lowering the temperature can be understood from the known negative thermal-expansion behavior and also from the fact that the high pressure phase can also be approached at ambient pressure by cooling. In order to understand the M₃ phonon softening in ReO₃ we have done lattice-dynamical calculations that support the qualitative arguments given above.