3564 Talthybius is a Jupiter Trojan asteroid that orbits in the L4 Lagrangian point of the Sun-Jupiter system, in the "Greek Camp" of Trojan asteroids. It was named after the Greek hero Talthybius, who was a herald during the Trojan War. It was discovered by Edward L. G. Bowell on October 15, 1985 in Flagstaff, Arizona at the Anderson Mesa station of the Lowell Observatory.