  The significant divergence between the SM predictions and experimental
measurements for the ratios, $R_{D^{(*)}}\equiv \mathcal{B}(\bar{B}\to
D^{(*)}\tau^- \bar{\nu}_\tau)/ \mathcal{B}(\bar{B}\to D^{(*)} \ell^{\prime-}
\bar{\nu}_{\ell^{\prime}})$ with $(\ell^{\prime}=e\,,\mu)$, implies possible
hint of new physics in the flavour sector. In this paper, motivated by the
"$R_{D^{(*)}}$ puzzle" and abundant $B^*$ data samples at high-luminosity
heavy-flavor experiments in the future, we try to probe possible effects of new
physics in the semileptonic $\bar{B}^*_{u,d,s} \to P \ell^- \bar{\nu}_\ell$
$(P=D\,,D_s\,,\pi\,,K)$ decays induced by $b \to (u,\,c)\ell^-
\bar{\nu}_{\ell}$ transitions in the model-independent vector and scalar
scenarios. Using the spaces of NP parameters obtained by fitting to the data of
$R_{D}$ and $R_{D^{*}}$, the NP effects on the observables including branching
fraction, ratio $R_P^{\ast}$, lepton spin asymmetry and lepton forward-backward
asymmetry are studied in detail. We find that the vector type couplings have
large effects on the branching fraction and ratio $R_P^{\ast}$. Meanwhile, the
scalar type couplings provide significant contributions to all of the
observables. The future measurements of these observables in the $\bar{B}^* \to
P \ell^- \bar{\nu}_\ell$ decays at the LHCb and Belle-II could provide a way to
crosscheck the various NP solutions to the "$R_{D^{(*)}}$ puzzle".
