In this work we consider ${\mathrm{AdS}}_3$ gravitational theory with certain mixed boundary conditions at infinity. Using the Chern-Simons formalism of ${\mathrm{AdS}}_3$ gravity, we find that these mixed boundary conditions lead to nontrivial boundary terms, which, in turn, produce exactly the spectrum of the $T\overline{T}/J\overline{T}$-deformed conformal field theories (CFTs). We then follow the procedure for constructing asymptotic boundary dynamics of ${\mathrm{AdS}}_3$ to derive the constrained $T\overline{T}$-deformed Wess-Zumino-Witten (WZW) model from Chern-Simons gravity. The resulting theory turns out to be the $T\overline{T}$-deformed Alekseev-Shatashvili action after disentangling the constraints. Furthermore, by adding a $U\left(1\right)$ gauge field associated to the current $J$, we obtain one type of the $J\overline{T}$-deformed WZW model, and show that its action can also be constructed from the gravity side. These results provide a check on the correspondence between the $T\overline{T}/J\overline{T}$-deformed CFTs and the deformations of boundary conditions of ${\mathrm{AdS}}_3$, the latter of which may be regarded as coordinate transformations.