Bloom's father Rudolph apparently named his dog "Athos" after the character in Alexandre Dumas' The Three Musketeers, but there also seems to be an echo of Odysseus' dog, who has the very similar name of Argos. Argos perks up and then dies after seeing Odysseus, who left for Troy 19 years earlier. Bloom thinks in Hades that Athos too "took it to heart, pined away" when his master died.
In Book 17 of the Odyssey Eumaeus and Odysseus pass by an impressive-looking old dog lying in piles of dung and covered with fleas. The dog raises his head, pricks up his ears, wags his tails, and then loses the strength to hold his ears up, much less to approach his old master. The disguised Odysseus wipes away a tear and questions Eumaeus about the handsome animal. After telling the beggar about the dog's glory days, the noble man who raised him from a puppy, and the duties that are forgotten when a lord goes away, Eumaeus goes into the palace to tend to the needs of the insolent suitors. Argos, having finally seen the man he longed for, gives up the ghost.
Bloom, who is kind to stray dogs (in Circe he gives one the meat he has bought, and in Eumaeus he remembers Molly's anger when he tried to bring one home), remembers his father's instructions in his suicide note: "Be good to Athos, Leopold, is my last wish." He recalls the note again in Ithaca: "be kind to Athos, Leopold...," remembering Athos as "an infirm dog."