In the present work, extended X-ray absorption fine-structure (EXAFS) investigations of Co₆₉Fe _x Si_21–x B₁₀ (x = 3, 5, 7) glassy ribbons were performed at the Co K-edge. The magnitude of the first peak of the Fourier transforms of the EXAFS signals is found to increase monotonically with increasing Si concentrations indicating the formation of the localized ordered structure at the atomic scale. The Co–Si coordination number (CN) increases at the expense of the CN of Co/Fe. Smaller interatomic distances are observed in the glassy phase compared with that in the crystalline phase which promotes the stability of the glassy phase. Calculations of the thermodynamic parameter (P _HSS), cohesive energy (E _C) and the atomic radius difference (δ) parameter show that the alloy composition Co₆₉Fe₃Si₁₈B₁₀ has a good glass-forming ability (GFA) with the highest CN of Si compared with other compositions. A linear correlation of CN with that of the GFA parameter (P _HSS) exists and the CN also plays a crucial role in the GFA of the glassy alloys. This parameter should be considered in developing different GFA criteria.