The group's foreign minister made the request in a letter on Monday. A UN committee will rule on the request.
The Taliban also nominated their Doha-based spokesperson, Suhail Shaheen, as Afghanistan's UN ambassador.
The group, which seized control of Afghanistan last month, said the envoy for the ousted government no longer represented the country.
The request to participate in the high-level debate is being considered by a credentials committee, whose nine members include the US, China and Russia, according to a UN spokesperson.