I think the word покрывать is a very close equivalent. Seems to be applicable to all four-legged animals. See here and here.
Пристраиваться as suggested by jwalker is another option, though the exact meaning may be unclear depending on the situation. Пристроиться may very well mean just a cosy comfortable position. To achieve the sexual connotation, one would have to make sure to use it with к (to emphasize the direction) rather than на, у (location). So:
пристроитьсякегоноге
rather than
пристроитьсяуегоноги.
The reason why there is no clarity in this topic is probably that all sex-related topics in general have long been a taboo in the USSR. The same explains why the words used to describe sex to a large extent belong to profanity in Russian.