African grays are pretty smart. I'm not sure I'd go so far as to call them near-human, but from what I've read there's a case for putting them on par with cetaceans or nonhuman primates.

The real trouble is that the research into this sort of thing is fiendishly subjective and surprisingly sparse. Even a detailed ordering of relative animal intelligence involves a lot of decisions about which researchers to trust, and comparison with humans is worse.