In this work, we have studied the $D_s^+$ decays into $\pi^+\pi^+\pi^-$ and $\pi^+K^+K^-$ mesons using the techniques from chiral unitary approach to account for the final state interactions in those processes. For this we have adopted a model in which the $D_s^+$ meson decays weakly into a $\pi^+$ and a quark-antiquark pair $q\bar{q}$, which after hadronization produces two pseudoscalar mesons that undergo final state interactions. As a result, we obtain the invariant mass distributions associated with the pairs $\pi^+\pi^-$ and $K^+K^-$, which proceed through $f_0(980)$ decay, and compare them with that ones available from the experiment. Our findings are in a fair agreement with the shape of these experimental data, where a $f_0(980)$ signal is seen in both $\pi^+\pi^-$ and $K^+K^-$ distributions. In addition, we have predicted the relative size of these distributions which can be measured by the experimental facilities.