We benefit from the lattice QCD determination by the HPQCD of the Standard Model (SM) form factors for the ${\overline{B}}_s\to D_s$ $$ {\overline{B}}_s\to {D}_s $$ [Phys. Rev. D 101 (2020) 074513] and the SM and tensor ones for the ${\overline{B}}_s\to D_s^{\ast }$ $$ {\overline{B}}_s\to {D}_s^{\ast } $$ ( arXiv:2304.03137 [hep-lat]) semileptonic decays, and the heavy quark effective theory (HQET) relations for the analogous B → D (*) decays obtained by F.U. Bernlochner et al. in Phys. Rev. D 95 (2017) 115008, to extract the leading and sub-leading Isgur-Wise functions for the ${\overline{B}}_s\to D_s^{\left(\ast \right)}$ $$ {\overline{B}}_s\to {D}_s^{\left(\ast \right)} $$ decays. Further use of the HQET relations allows us to evaluate the corresponding scalar, pseudoscalar and tensor form factors needed for a phenomenological study of new physics (NP) effects on the ${\overline{B}}_s\to D_s^{\left(\ast \right)}$ $$ {\overline{B}}_s\to {D}_s^{\left(\ast \right)} $$ semileptonic decay. At present, the experimental values for the ratios $R_{D^{\left(\ast \right)}}=\Gamma \left(\overline{B}\to D^{\left(\ast \right)}{\tau }^{-}{\overline{\nu }}_{\tau }\left(/\Gamma \right)\overline{B}\to D^{\left(\ast \right)}{e}^{-}\left({\mu }^{-}\right){\overline{\nu }}_{e\left(\mu \right)}\right)$ $$ {\mathcal{R}}_{D^{\left(\ast \right)}}=\Gamma \left[\overline{B}\to {D}^{\left(\ast \right)}{\tau}^{-}{\overline{\nu}}_{\tau}\left]/\Gamma \right[\overline{B}\to {D}^{\left(\ast \right)}{e}^{-}\left({\mu}^{-}\right){\overline{\nu}}_{e\left(\mu \right)}\right] $$ are the best signal in favor of lepton flavor universality violation (LFUV) seen in charged current (CC) b → c decays. In this work we conduct a study of NP effects on the ${\overline{B}}_s\to D_s^{\left(\ast \right)}{\tau }^{-}{\overline{\nu }}_{\tau }$ $$ {\overline{B}}_s\to {D}_s^{\left(\ast \right)}{\tau}^{-}{\overline{\nu}}_{\tau } $$ semileptonic decays by comparing tau spin, angular and spin-angular asymmetry distributions obtained within the SM and three different NP scenarios. As expected from SU(3) light-flavor symmetry, we get results close to the ones found in a similar analysis of the $\overline{B}\to D^{\left(\ast \right)}$ $$ \overline{B}\to {D}^{\left(\ast \right)} $$ case. The measurement of the ${\overline{B}}_s\to D_s^{\left(\ast \right)}\ell {\overline{\nu }}_{\ell }$ $$ {\overline{B}}_s\to {D}_s^{\left(\ast \right)}\ell {\overline{\nu}}_{\ell } $$ semileptonic decays, which is within reach of present experiments, could then be of relevance in helping to establish or rule out LFUV in CC b → c transitions.