"Regarding terrorism, we have made progress and may soon be in the position to conclude a draft text outlining the Taliban’s commitments to ensure Afghan soil is never again used for international terrorism", Hunter said on Wednesday.
The question of the presence of US troops in the country, which is condition-based, would also be soon discussed with the Taliban movement, Hunter noted.
"The United States has made clear to the Taliban that we are prepared to reduce our forces. However, we have not agreed to [a] number or timeline with the Taliban", he said.
According to the US envoy, the final disposition of forces will be determined with a peace settlement and with the establishment of a new government.
Hunter added that the United States is prepared to create a foundation for intra-Afghan talks, to begin as soon as possible.
The United States and the Taliban have held several rounds of talks and are preparing for the next summit with expectations of a breakthrough.
Taliban spokesman Suhail Shaheen earlier said that the US had agreed on troop withdrawals and that was the main outcome of talks to date.
The Taliban is a terrorist group banned in Russia and many other countries.