Most Atheists, I think, would deny the existence of a human "soul" in the Cartesian sense.
However, the defining characteristic of Atheism is disbelief in God or gods. As per Wikipedia:
Atheism, in the broadest sense, is an absence of belief in the existence of deities. Less broadly, atheism rejects the belief that any deities exist. In an even narrower sense, atheism is specifically the position that there are no deities.
In some sense, then, Atheism hinges on the denial of deities. I know a few people who don't explicitly believe in a God or gods but nevertheless would describe themselves as vaguely "spiritual" in some sense. I don't know what they mean by "spiritual" and I'm not sure that they do either, but it seems to suggest to me that they believe in some sort of immaterial "spirit" that is in addition to their bodily sack of meat.
I don't know anyone "spiritual" who actively denies the existence of God/gods, but here is one example of someone who seems to have something akin to this. At the very least, it seems hypothetically possible to at once believe that there is no God/gods, and simultaneously that there is an immaterial "soul".
On the other hand, someone who denies the existence of a soul might likely be an Atheist. This is not a necessity: it seems, for example, that some Christian thinkers rejected the existence of a "soul". I don't personally know of any examples on the top of my head, but I suspect that there have been those in other religions (particularly Deism) who might believe in a supernatural creator without believing in a human soul.
So, there are those who disbelieve in the existence of a human soul, including not only Atheists but also Christians and probably others. On the other hand, the majority of those in the 4 largest world religions seem to believe in the existence of a human soul, and some folks who disbelieve in God/gods are still "spiritual" in some sense.
Is there a term specifically for disbelief in the human "soul" or "spirit"? I.e. a term that would unequivocally indicate a belief that humans are not more than their material sack of meat, but that would not imply either belief or disbelief in God/gods?