The "language-ready " brain theory suggests that the infant brain is pre-wired for language acquisition prior to language exposure. As a potential brain marker of such a language readiness, a leftward structural brain asymmetry was found in human infants for the Planum Temporale (PT), which overlaps with Wernicke's area. In the present longitudinal in vivo MRI study conducted in 35 newborn monkeys (Papio anubis), we found a similar leftward PT surface asymmetry. Follow-up rescanning sessions on 29 juvenile baboons at 7-10 months showed that such an asymmetry increases across the two ages classes. These original findings in non-linguistic primate infants strongly question the idea that the early PT asymmetry constitutes a human infant-specific marker for language development. Such a shared early perisylvian organization provides additional support that PT asymmetry might be related to a lateralized system inherited from our last common ancestor with Old-World monkeys at least 25-35 million years ago.