  We discuss $B_c \to \eta_c$ and $B_c \to J/\psi$ semileptonic decays within
the Standard Model (SM) and beyond. The relevant transition form factors, being
the main source of theoretical uncertainties, are calculated in the sum rule
approach and are provided in a full $q^2$ range. We calculate the semileptonic
branching fractions and find for the ratios, $R_{\eta_c}|_{\rm SM}= 0.32 \pm
0.02$, $R_{J/\psi}|_{\rm SM} = 0.23 \pm 0.01$. Both predictions are in
agreement with other existing calculations and support the current tension in
$R_{J/\psi}$ at 2$\sigma$ level with the experiment. To extend the potential of
testing the SM in the semileptonic $B_c$ decays, we consider the
forward-backward asymmetry and polarization observables. We also study the
4-fold differential distributions of $B_c \rightarrow J/\psi (J/\psi \to
\tilde{\ell}^-\tilde{\ell}^+ ) \ell^- \bar{\nu}_\ell$, where $\tilde{\ell} = e,
\mu$, in the presence of different new physics scenarios and find that the new
physics effects can significantly modify the angular observables and can also
produce effects which do not exist in the SM. Using the constraints on the new
physics couplings from the recent combined analysis of BaBar, Belle and LHCb
data on semileptonic $B \to D^{(\ast)}$ decays, where the effects of new
physics could be visible, we find that these different new physics scenarios
are not able to simultaneously explain the current experimental value of
$R_{J/\psi}$.
