This question has stimulated a good deal of philosophical debate among scholars down the centuries. Nevertheless, the general consensus seems to be that Aristotle believed that the intellect was immortal. This is because, as Aristotle argues in De Anima, thinking isn't a function of a particular bodily organ, not even the brain. So when the body dies and decays, the intellect lives on, as it's the only living function of the body that isn't in any way related to a bodily organ.
Intellect must be distinguished here from the soul—which, according to Aristotle, must perish along with the body. In clear opposition to his mentor Plato, Aristotle argues that the soul, as the form or structure of the living body, cannot escape death. It is this intimate relation between the soul and the body that ensures that the soul cannot be immortal. The intellect, however, never dies, as it doesn't have the same degree of connection to the body as the soul.