  It has been shown that the anomalies observed in $\bar B\to
D^{(\ast)}\tau\bar\nu_\tau$ and $\bar B\to \bar K\ell^+\ell^-$ decays can be
resolved by adding a single scalar or vector leptoquark to the Standard Model,
while constraints from other precision measurements in the flavour sector can
be satisfied without fine-tuning. To further explore these two interesting
scenarios, in this paper, we study their effects in the semi-leptonic
$\Lambda_b\to\Lambda_c\tau\bar\nu_\tau$ decay. Using the best-fit solutions for
the operator coefficients allowed by the current data of mesonic decays, we
find that (i) the two scenarios give similar amounts of enhancements to the
branching fraction $\mathcal B(\Lambda_b\to\Lambda_c\tau\bar\nu_\tau)$ and the
ratio $R_{\Lambda_c}=\mathcal B(\Lambda_b\to\Lambda_c
\tau\bar\nu_\tau)/\mathcal B(\Lambda_b\to\Lambda_c\ell\bar\nu_\ell)$, (ii) the
two best-fit solutions in each of these two scenarios are also
indistinguishable from each other, (iii) both scenarios give nearly the same
predictions as those of the Standard Model for the longitudinal polarizations
of $\Lambda_c$ and $\tau$ as well as the lepton-side forward-backward
asymmetry. With future measurements of these observables in
$\Lambda_b\to\Lambda_c\tau\bar\nu_\tau$ decay at the LHCb, the two leptoquark
scenarios could be further tested, and even differentiated from the other NP
explanations for the $R_{D^{(\ast)}}$ anomalies. We also discuss the
feasibility for the measurements of these observables at the LHC and the future
$e^+e^-$ colliders.
