Using a combination of scanning tunneling spectroscopy and atomic lateral manipulation, we obtained a systematic variation of the Kondo temperature T_K of Co atoms on Ag(111) as a function of the surface-state contribution to the total density of states at the atom adsorption site ρ_s. By sampling the T_K of a Co atom on positions where ρ_s was spatially resolved beforehand, we obtain a nearly linear relationship between the magnitudes. We interpret the data on the basis of an Anderson model including orbital and spin degrees of freedom [SU(4)] in good agreement with the experimental findings. The fact that the onset of the surface band is near the Fermi level is crucial to finding the observed linear behavior. In light of this model, the quantitative analysis of the experimental data evidences that at least a quarter of the coupling of Co impurities with extended states takes place through the hybridization to surface states. This result is of fundamental relevance in the understanding of Kondo screening of magnetic impurities on noble-metal surfaces, where bulk and surface electronic states coexist.