Organic thiols are known to react with gold surface to form self-assembled monolayers (SAMs), which can be used to produce materials with highly attractive properties. Although the structure of various SAMs is widely investigated, some aspects of their formation still represent a matter of debate. One of these aspects is the mechanism of S-H bond dissociation in thiols upon interaction with gold. This work presents a new suggestion for this mechanism on the basis of DFT study of methanethiol interaction with a single gold atom and a Au(20) cluster. The reaction path of dissociation is found to be qualitatively independent of the model employed. However, the highest activation barrier of S-H bond dissociation on the single gold atom (12.9 kcal/mol) is considerably lower than that on the Au(20) cluster (28.9 kcal/mol), which can be attributed to the higher extent of gold unsaturation. The energy barrier of S-H cleavage decreases by 4.6 kcal/mol in the presence of the second methanethiol molecule at the same adsorption site on the model gold atom. In the case of the Au(20) cluster we have observed the phenomenon of hydrogen transfer from one methanethiol molecule to another, which allows reducing the energy barrier of dissociation by 9.1 kcal/mol. This indicates the possibility of the "relay" hydrogen transfer to be the key step of the thiol adsorption observed for the SAMs systems.