Afghan President Ashraf Ghani fled the country Sunday after Taliban fighters entered Kabul, the nation's capital, demanding a transfer of power.
Taliban spokesman Suhail Shaheen told the Al-Jazeera English news channel in Qatar about the latest development.
The insurgents were directed to refrain from violence and offer safe passage to anyone wishing to leave the capital.
"No one's life, property, and dignity will be harmed and the lives of the citizens of Kabul will not be at risk," the group said in a statement also warning that no one should enter the area around the capital.
In just over a week, fighters have taken over nearly all of Afghanistan as thousands of people have fled the war-torn country.
A duty officer with the Defense Press Operations confirmed that roughly 3,000 American troops were heading to Kabul on Saturday to assist with the evacuation of U.S. civilian personnel.
Two Marine Corp infantry battalions and one Army infantry battalion joined the 1,000 troops already at the airport and the embassy.
Last week, 600 British troops were being deployed to Kabul to assist with a safe evacuation of 3,000 British nationals and 2,000 Afghans who worked with British troops.
The insurgents also seized Maidan Shar, the capital of Maidan Wardak, and the land border at Torkham on Sunday.
During a televised speech on Saturday, Ghani addressed the country, promising not to abandon the "achievements" they gained since the U.S. upset the Taliban following the Sept. 11 attacks.
"We have started consultations, inside the government with elders and political leaders, representatives of different levels of the community as well as our international allies," the president said.