There was no confirmation from the Afghan government about the Taliban’s claim.
In addition, the militants entered multiple districts in the Afghan capital.
“Military units of the Islamic Emirate of Afghanistan have entered Kabul city to ensure security,” Taliban spokesman Zabihullah Mujahid tweeted, adding that “their advance is continuing normally”.
The insurgents said they were seeking complete power.
Two officials from the Taliban told Reuters there would be no transitional government following their lighting sweep across Afghanistan that led back to the capital two decades after the group were overthrown by US-led forces.
American diplomats were evacuated from their embassy by helicopter to the airport as local Afghan forces, trained for years and equipped by the United States and others for billions of dollars, melted away.
Ghani’s destination was uncertain: a senior Interior Ministry official said he had left for Tajikistan, while a Foreign Ministry official said his location was unknown and the Taliban said it was checking his whereabouts.
Some local social media users branded him a “coward” for leaving them in chaos.
Taliban fighters reached Kabul “from all sides”, the senior Interior Ministry official told Reuters and there were some reports of sporadic gunfire around the city.
A Kabul hospital said more than 40 people wounded in clashes on the outskirts were being treated, but there did not appear to be major fighting.
It was not clear yet how power would be transferred.
The Taliban said they were waiting for the Western-backed government to surrender peacefully.
“Taliban fighters are to be on standby on all entrances of Kabul until a peaceful and satisfactory transfer of power is agreed,” said spokesman Zabihullah Mujahid.
The government’s acting interior minister, Abdul Sattar Mirzakawal, said power would be handed over to a transitional administration.
“There won’t be an attack on the city, it is agreed that there will be a peaceful handover,” he tweeted.
A Taliban spokesman said that Taliban fighters have entered Kabul in response to a “law and order issue”, the Guardian reported.
The group ordered its fighters earlier Sunday not to enter the capital, saying the remnants of the government’s forces were responsible for security.
But later, a spokesman tweeted that Taliban forces should enter areas deserted by Afghan forces in order to maintain law and order.
The Taliban issued a statement under the “Islamic Emirate of Afghanistan” banner saying the group has now been permitted to enter Kabul.
The statement claimed that Afghan police and other relevant institutions abandoned their duties and that to “prevent theft, looting and crime”, the group’s forces have been allowed to enter the capital.
“The Taliban will secure areas abandoned by Afghan forces in Kabul,” said the statement.
It sought to assure citizens that the forces will neither enter their homes, nor “bother them”.
Journalists reporting from on the ground, including former Wall Street Journal reporter Habib Khan, confirmed that the Taliban will enter the city to secure areas abandoned by the government in order to “control the chaos”.
In other news, the US Embassy in Kabul said in a security alert that the security situation in the Afghan capital was changing quickly, including at the airport, where there were reports of gunfire as US troops aid an evacuation of most US personnel.
“There are reports of the airport taking fire; therefore we are instructing U.S. citizens to shelter in place,” the embassy said.
The Guardian also reported a Kabul Hospital tweeting that “more than 40 people” have been wounded in clashes on the outskirts of Kabul and are receiving treatment there.
Meanwhile, the CNN reported that a high-level Afghan government delegation will travel to Doha “soon” for talks with the Taliban.
“The situation is changing by the minute but we could expect an Afghan government delegation that has more power and authority to travel to Doha soon,” a source privy to the intra-Afghan negotiations told CNN.
A spokesman for the Taliban told the BBC the group want to take control of Afghanistan “in the next few days”, as their fighters encircled Kabul, the capital.
“In next few days, we want a peaceful transfer,” Suhail Shaheen, based in Qatar as part of the group’s negotiating team, told the BBC.
Shaheen laid out the policies of the Taliban ahead of an expected power transfer that would re-install the group two decades after US-led forces toppled them in the wake of the September 11, 2001 attacks.
“We want an inclusive Islamic government… that means all Afghans will be part of that government,” Shaheen said.
“We will see that in the future as the peaceful transfer is taking place.”
He also said foreign embassies and workers would not be targeted by the group’s fighters and they should remain in the country.
“There will be no risk to diplomats, NGOs, to anyone. All should continue their work as they were continuing in the past. They won’t harm them, they should remain.”
Rebuffing fears the country would be plunged back to the dark days of the group’s ultra-conservative version of Islamic law, Shaheen said the Taliban will instead seek a “new chapter” of tolerance.
“We want to work with any Afghan, we want to open a new chapter of peace, tolerance, peaceful coexistence and national unity for the country and for the people of Afghanistan,” he said.
But many officials, soldiers and police have surrendered or abandoned their posts, fearing reprisals against anyone suspected of working with the Western-backed government or Western forces.
“We reassure that there is no revenge on anyone. Any case will be investigated.”
The Doha-based spokesman said the group would also review its relationship with the United States, which it has waged a deadly insurgency against for decades.
“Our relationship was in the past,” he said. “In future, if it will touch our agenda no more, it will be a new chapter of cooperation.”