We study the $\bar B_s^0 \to J/\psi f_0(980)$ and $\bar B_s^0 \to J/\psi a_0(980)$ reactions, and pay attention to the different sources of isospin violation and mixing of $f_0(980)$ and $a_0(980)$ resonances where these resonances are dynamically generated from meson-meson interaction. We find that the main reason of isospin violation is the isospin breaking in the meson-meson transition $T$ matrices, and the other source is that the loops involving kaons in the production mechanism provide a source of isospin violation because they do not cancel by the different mass of the charged and neutral kaons. We obtain a branching ratio for $a_0(980)$ production of the order of $5 \times 10^{-6}$. Future experiments can address this problem and the production rate and shape of the $\pi^0 \eta$ mass distribution will definitely help to better understand the nature of scalar resonances. Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Article funded by SCOAP3 and published under licence by Chinese Physical Society and the Institute of High Energy Physics of the Chinese Academy of Science and the Institute of Modern Physics of the Chinese Academy of Sciences and IOP Publishing Ltd.