  Polarization measurements in $\bar{B} \to D^{(\ast)} \tau \overline{\nu}$ are
useful to check consistency in new physics explanations for the $R_{D}$ and
$R_{D^{\ast}}$ anomalies. In this paper, we investigate the $D^{\ast}$ and
$\tau$ polarizations and focus on the new physics contributions to the fraction
of a longitudinal $D^{\ast}$ polarization ($F_{L}^{D^{\ast}}$), which is
recently measured by the Belle collaboration $F_{L}^{D^{\ast}} = 0.60 \pm
0.09$, in model-independent manner and in each single leptoquark model (${\rm
R}_2$, ${\rm S}_1$ and ${\rm U}_1$) that can naturally explain the
$R_{D^{(\ast)}}$ anomalies. It is found that $\mathcal{B}(B_c^{+} \to \tau^{+}
\nu)$ severely restricts deviation from the Standard Model (SM) prediction of
$F_{L, \textrm{SM}}^{D^{\ast}} = 0.46 \pm 0.04$ in the leptoquark models:
[0.43, 0.44], [0.42, 0.48], and [0.43, 0.47] are predicted as a range of
$F_{L}^{D^{\ast}}$ for the ${\rm R}_2$, ${\rm S}_1$, and ${\rm U}_1$ leptoquark
models, respectively, where the current data of $R_{D^{(\ast)}}$ is satisfied
at $1\,\sigma$ level. It is also shown that the $\tau$ polarization observables
can much deviate from the SM predictions. The Belle II experiment, therefore,
can check such correlations between $R_{D^{(\ast)}}$ and the polarization
observables, and discriminate among the leptoquark models.
