We have phenomenologically investigated the decays $B_s^0\to X\left(3872\right){\pi }^{+}{\pi }^{-}\left(K^{+}{K}^{-}\right)$ and $B_s^0\to \psi \left(2S\right){\pi }^{+}{\pi }^{-}\left(K^{+}{K}^{-}\right)$. In our analysis, the scalar meson $f_0\left(980\right)$ is formed through the final state interactions of coupled channels $\pi \pi$ and $K\overline{K}$. Our findings indicate that the ${\pi }^{+}{\pi }^{-}$ invariant mass distribution of the $B_s^0\to \psi \left(2S\right){\pi }^{+}{\pi }^{-}$ decay can be accurately reproduced. Furthermore, we have explored the ${\pi }^{+}{\pi }^{-}\left(K^{+}{K}^{-}\right)$ invariant mass distribution of the $B_s^0\to X\left(3872\right){\pi }^{+}{\pi }^{-}\left(K^{+}{K}^{-}\right)$ decay, accounting for the different production mechanisms between $X\left(3872\right)$ and $\psi \left(2S\right)$, up to a global factor. It is found that the production rates for $X\left(3872\right)$ and $\psi \left(2S\right)$ are much different, which indicates that the structure of $X\left(3872\right)$ is more complicated than the $\psi \left(2S\right)$, which is a conventional $c\overline{c}$ state. Additionally, we have considered the contributions from $f_0\left(1500\right)$ to ${\pi }^{+}{\pi }^{-}$ and the $\varphi$ meson to $K^{+}{K}^{-}$ in our analysis. Utilizing the model parameters, we have calculated the branching fraction of $B_s^0\to X\left(3872\right)K^{+}{K}^{-}$, and anticipate that the findings of our study can be experimentally tested in the future.