We have characterized the short-range order in the liquid and undercooled states of Au-Si alloy at the eutectic composition using molecular dynamics simulations. The interactions are described via a modified embedded-atom model refined to take into account the liquid properties. For the eutectic liquid, the local structure is characterized by a strong Au-Si affinity, namely a well-pronounced chemical short-range order which leads to the slowing down of the formation of icosahedral local motifs in the undercooled regime. Moreover we discuss the influence of this peculiar local structure on the dynamic and thermodynamic properties of the liquid phase and compare our results with available experimental data.