  Superconducting transition temperature $T_C$ of some of the cubic
$\beta$-phase Mo$_{1-x}$Re$_x$ alloys with x > 0.10 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 ($\rho$(T)) and
heat capacity (C$_P$(T)), as well as the theoretical estimation of electronic
density of states (DOS) using band structure calculations. The $\rho$(T) in the
normal state of the Mo$_{1-x}$Re$_x$ alloys with x > 0.15 is distinctly
different from that of Mo and the alloys with x < 0.10. We have also observed
that the Sommerfeld coefficient of electronic heat capacity $\gamma$,
superconducting transition temperature $T_C$ and the DOS at the Fermi level
show an abrupt change above x > 0.10. The analysis of these results indicates
that the value of electron-phonon coupling constant {\lambda}ep required to
explain the $T_C$ of the alloys with x > 0.10 is much higher than that
estimated from $\gamma$. On the other hand the analysis of the results of the
$\rho$(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 $T_C$ in the
$\beta$-phase Mo$_{1-x}$Re$_x$ alloys with x > 0.10.
