Superconducting transition temperature TC of some of the cubic β phase Rex alloys with is an order of magnitude higher than that in the elements Mo and Re. We investigate this rather enigmatic issue of the enhanced superconductivity with the help of experimental studies of the temperature dependent electrical resistivity (ρ(T)) and heat capacity (CP(T)), as well as the theoretical estimation of electronic density of states (DOS) using band structure calculations. The ρ(T) in the normal state of the Rex alloys with is distinctly different from that of Mo and the alloys with We have also observed that the Sommerfeld coefficient of electronic heat capacity γ, superconducting transition temperature TC and the DOS at the Fermi level show an abrupt change above . The analysis of these results indicates that the value of electron–phonon coupling constant λep required to explain the TC of the alloys with is much higher than that estimated from γ. On the other hand the analysis of the results of the ρ(T) reveals the presence of phonon assisted inter-band s–d scattering in this composition range. We argue that a strong electron–phonon coupling arising due to the multiband effects is responsible for the enhanced TC in the β phase Rex alloys with .