  We reconsider the possibility that the tension in the $|V_{cb}|$
determinations from inclusive and exclusive $B$ decay modes is due to a new
physics effect. We modify the Standard Model effective Hamiltonian for
semileptonic $b \to c$ transitions including a tensor operator with a lepton
flavour dependent coupling $\epsilon_T^\ell$, and investigate separately the
muon and electron modes. The interference term between SM and NP, proportional
to the lepton mass, has different impact in the inclusive and exclusive $B$
modes to muon. Moreover, even when the lepton mass is small as for the
electron, the NP effect is different in inclusive and exclusive $B$ channels.
For both $\mu$ and $e$ we find a region of $\epsilon_T^{\mu,\,e}$ where the
constraints from $B^- \to D^{(*)0} \ell^- {\bar \nu}_\ell$ and $B \to X_c \,
\ell \, \bar \nu_\ell$ are satisfied for the same $|V_{cb}|$.
