A three Higgs-doublet model exhibiting $S_3$ -symmetry can predict the observed pattern of quark masses and their mixings. However, the same symmetry also introduces potential flavor-changing neutral currents (FCNCs) at the tree level. In this study, we assume that the scalar potential contains appropriate soft $S_3$ -breaking terms to maintain flexible choices of scalar masses. We identify the parameters in the Yukawa Lagrangian of the quark sector responsible for such FCNCs and constrain them using data from flavor physics observables, such as meson-decays and meson-mixings. We also validate the corresponding model parameter space via renormalization group evaluation.