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.
But they are unlikely to meet before the end of the General Assembly session next Monday. Until then, under UN rules, Ghulam Isaczai will remain Afghanistan’s ambassador to the global body.
They also said that several countries no longer recognised former President Ashraf Ghani as leader.
When the Taliban last controlled Afghanistan, between 1996 and 2001, the ambassador of the government they overthrew stayed on as a UN representative, after the credentials committee deferred its decision on competing claims for the position.
At the UN meeting on Tuesday, Qatar urged world leaders to stay engaged with the Taliban.
“Boycotting them would only lead to polarisation and reactions, whereas dialogue could be fruitful,” said Qatar’s ruler, Sheikh Tamim bin Hamad Al Thani.
- IN PICTURES: Ten days that shook Afghanistan
- EXPLAINER: Who’s who in the Taliban leadership?
- ANALYSIS: What has changed in Afghanistan in 20 years
The country has helped Afghans and foreign nationals to evacuate the country since the Taliban takeover, and has facilitated recent intra-Afghan peace talks.