Co single atom junctions on copper surfaces are studied by scanning tunneling microscopy and ab initio calculations. The Kondo temperature of single cobalt atoms on the Cu(111) surface has been measured at various tip-sample distances ranging from tunneling to the point contact regime. The experiments show a constant Kondo temperature for a whole range of tip-substrate distances consistently with the predicted energy position of the spin-polarized d levels of Co. This is in striking difference to experiments on Co/Cu(100) junctions, where a substantial increase of the Kondo temperature has been found. Our calculations reveal that the different behavior of the Co adatoms on the two Cu surfaces originates from the interplay between the structural relaxations and the electronic properties in the near-contact regime.