A few studies of alkali atoms deposition on the (110)-rutile surface have been reported in the last years. The process involves a selective electron transfer from the alkali atoms to the surface penta-fold Ti atoms, and it has been well characterized by both experimental and theoretical techniques. However, the adsorption site of the alkali atoms on the surface is still unclear and different models have been proposed. In our previous works we omitted some aspects which have been considered in this study. In particular, we have examined the role of the relaxation of bridging oxygen atoms and those factors responsible for adsorption structure from ab initio calculations and molecular dynamics simulations. The results indicate that there exist two characteristic adsorption sites described as "between" and "adjacent" sites. At very low coverages, in agreement with our previous work, alkali metal atoms do only occupy "between" sites binding to two bridging oxygen atoms. However, the number of alkali atoms in "adjacent" sites, at which the adsorbate binds to one bridging oxygen atom, increases as coverage does and as ionic radius decreases, but even at high coverages a nonnegligible percentage of alkali atoms in "between" sites is still present.