In this work we study an extension of the $SU(5)$ grand unification model in the supersymmetric case, which consists in adding a $S_3$-modular symmetry. This symmetry takes the role of a flavour symmetry between quarks. Futhermore, the Higgs scalar sector is extended to include six doublets in contrast to the minimal Higgs sector of the Standard Model of Elementary Particles. Then we proceed to group the particle fields in the different irreducible representations of $SU(5)$ and within the irreducible representations of $S_3$. Once the matter content of the model is established, we proceed to construct the Yukawa potential where modular forms are used and specific modular weights are assigned to the particle fields in order to make the potential invariant under the flavour symmetry. From the mass matrices associated with the potential, which exhibit NNI textures, the CKM mixing matrix is ​​constructed. In view of these results and by means of a $\chi ^2$ adjustment, we seek to reproduce the mass hierarchy and the mixing angles between the different quark flavors that are known experimentally on the $M_Z$ scale.