  Most of the popular explanations of the observed anomalies in the
semileptonic $B$-meson decays involve TeV scale Leptoquarks (LQs). Among the
various possible LQ models, two particular LQs -- $S_{1}(3, 1, 1/3)$ and
$U_{1}(3, 1, 2/3)$ seem to be most promising. Here, we use current LHC data to
constrain the $S_{1}(3, 1, 1/3)$ and $U_{1}(3, 1, 2/3)$ parameter spaces
relevant for the $R_{D^{(*)}}$ observables. We recast the latest ATLAS
$\tau\tau$ resonance search data to obtain new exclusion limits. For this
purpose, we consider both resonant (pair and single productions) and
non-resonant ($t$-channel LQ exchange) productions of these LQs at the LHC. For
the limits, the most dominant contribution comes from the (destructive)
interference of the non-resonant production with Standard Model backgrounds.
The combined contribution from the pair and inclusive single production
processes is less prominent but non-negligible. The limits we get are
independent and competitive to other known bounds. For both the models, we set
limits on $R_{D^{(*)}}$ motivated couplings.
