A model for a site adsorption of an associating fluid on the crystalline surfaces is studied. The (100) and (111) lattice surfaces are considered. An integral equation method is used to obtain the density profiles and investigate adsorption. It is shown that the associative interaction between the bulk fluid particles and reactive atoms of the crystalline surface leads to a formation of surface complexes. They include monomeric and dimeric species of the bulk fluid and the atoms of the exposed solid plane. Adsorption on the on-top, 2-fold bridging site and the 4-fold hollow site positions is studied for the case of the (100) lattice surface. The complexes, which include one, two, and three surface atoms and bulk species, are investigated for (111) lattice surface.