The White House said Tuesday the Taliban had promised that civilians could travel safely to the Kabul Airport as the US military stepped up its airlift for Americans and Afghans fleeing the country.

Some 3,200 people have been evacuated by the US military so far, a White House official said, including 1,100 on Tuesday alone -- US citizens, permanent residents and their families on 13 flights.

Earlier Tuesday, the White House had said that 11,000 US nationals remain inside the country, including diplomats, contractors and others, most waiting to be evacuated after the Taliban takeover.

Washington wants to complete the exodus before its August 31 withdrawal deadline, and thousands of US soldiers were at the airport as the Pentagon planned to ramp up flights of its huge C-17 transport jets to as many as two dozen a day.

"Now that we have established the flow, we expect those numbers to escalate," said the White House official who gave the updated evacuation figures Tuesday and who spoke on condition of anonymity.

US officials said they were in contact with Taliban commanders to ensure the flight operations at Hamid Karzai International Airport remained safe from attack and that citizens and Afghans seeking to leave had safe passage.

"We have had no hostile interactions, no attack and no threat by the Taliban," said Major General Hank Taylor at the Pentagon.

But State Department spokesman Ned Price said Tuesday that the United States could decide to keep its core diplomatic presence, now operating out of the airport after the US embassy was shuttered, after August 31.

"If it is safe and responsible for us to potentially stay longer, that is something we may be able to look at," Price said.

He also called on the Taliban to follow through on promises to respect the rights of citizens including women.

Also being airlifted are Afghans granted US refugee visas, mostly for having worked as translators for American and NATO forces, other foreign nationals, and other unspecified "at risk" Afghans.

- Taliban assurances -

Despite some reports that people are being harassed and even beaten as they try to leave, White House National Security Advisor Jake Sullivan said "large numbers" had been able to reach the airport. "The Taliban have informed us they are prepared to provide the safe passage of civilians to the airport, and we intend to hold them to that commitment," he told reporters.

The US Department of Defence has poured troops into the airport since Saturday to protect the exodus as the Taliban insurgents entered Kabul after a lightning siege across the country and seized power.

Taylor said the number of US troops would rise from 2,500 on Monday to around 4,000 by late Tuesday. He said the United States aimed to increase its airlift to one plane an hour so that between 5,000 and 9,000 passengers could be carried out per day.

Some other countries, including Germany and France, have also been able to pick up their nationals and Afghans qualified to travel to those countries.

- Airport chaos -

Taylor spoke a day after security broke down at the airport, with videos showing hundreds of Afghans running next to a C-17, some clinging to it.

Videos appeared to show two people falling to their deaths from one aircraft after it took off. Another person was later found dead in a wheel well.

US Air Force spokeswoman Ann Stefanek said they were investigating the incidents. "Before the aircrew could offload the cargo, the aircraft was surrounded by hundreds of Afghan civilians," she said.

"Faced with a rapidly deteriorating security situation around the aircraft, the C-17 crew decided to depart the airfield as quickly as possible."

G7 summit on Afghanistan

US President Joe Biden and British Prime Minister Boris Johnson discussed Afghanistan on Tuesday and announced a virtual summit of the G7 leaders on the crisis, the White House said.

"They agreed to hold a virtual G7 leaders' meeting next week to discuss a common strategy and approach," the White House said in a statement.

This was the first phone call between Biden and a foreign leader since the startling weekend takeover by the Taliban of Kabul, prompting a panicky operation to withdraw final US and allied personnel from the city's airport.

The sudden Taliban victory has sparked fears of a large-scale humanitarian crisis both in Afghanistan and possibly involving waves of refugees seeking asylum abroad, including in western Europe.

Biden -- widely criticized for the lack of preparation in getting thousands of people airlifted to safety -- and Johnson "discussed the need for continued close coordination among allies and democratic partners on Afghanistan policy going forward," the statement said.

This includes "ways the global community can provide further humanitarian assistance and support for refugees and other vulnerable Afghans."

In London, a Downing Street spokesman said the two leaders welcomed US-British cooperation in the ongoing evacuation effort. 

"They resolved to continue working closely together on this in the days and weeks ahead to allow as many people as possible to leave the country," a statement said.

"The prime minister and President Biden agreed on the need for the global community to come together to prevent a humanitarian crisis," the statement said.

It said that Johnson also "stressed the importance of not losing the gains made in Afghanistan over the last twenty years."

The G7, which Britain heads this year, comprises Britain, Canada, France, Germany, Italy, Japan and the United States.

Taliban say won't seek 'revenge'

The Taliban have pledged not to seek "revenge" against their opponents in Afghanistan in their first press conference since taking power, as the United States said they would hold the insurgents to their promises to respect human rights.

The Taliban announcements came Tuesday after the return to Afghanistan of their co-founder Mullah Abdul Ghani Baradar, crowning the group's astonishing comeback after being ousted in a US-led invasion nearly 20 years ago.

In the capital Kabul, some shops opened and the insurgents told government staff to return to work -- though residents reacted cautiously and few women took to the streets.

Tens of thousands of people have tried to flee the country to escape the hardline Islamist rule expected under the Taliban, or fearing direct retribution for siding with the Western-backed government in power for the past two decades.

But Taliban spokesman Zabihullah Mujahid told reporters the new regime would be "positively different" from their 1996-2001 stint at the helm, infamous for deaths by stoning and barring women from working in contact with men.

"If the question is based on ideology, and beliefs, there is no difference... but if we calculate it based on experience, maturity, and insight, no doubt there are many differences," Mujahid told reporters.

"All those in the opposite side are pardoned from A to Z," he said. "We will not seek revenge."

Mujahid said a government would soon be formed but offered few details, only saying the Taliban would "connect with all sides".

He also said they were "committed to letting women work in accordance with the principles of Islam", without offering specifics.

A spokesman for the group in Doha, Suhail Shaheen, told Britain's Sky News that women would not be required to wear the all-covering burqa, but did not say what attire would be acceptable.

After the press conference, US State Department spokesman Ned Price said: "If the Taliban says they are going to respect the rights of their citizens, we will be looking for them to uphold that statement and make good on that statement."

- Triumphant return -

Baradar, now deputy leader of the Taliban, chose to touch down in Afghanistan's second biggest city Kandahar -- the Taliban's spiritual birthplace and capital during their first time in power.

He arrived from Qatar, where he has spent months leading talks with the United States and then Afghan peace negotiators. 

Before he landed, evacuation flights from Kabul's airport had resumed following a day of chaos at the facility, when huge crowds mobbed the tarmac.

Some people were so desperate to leave that they clung to the fuselage of a US military plane as it rolled down the runway for take-off. The US military said it was investigating deaths associated with that flight, including human remains found in a wheel well.

Washington has authorised the deployment of 6,000 troops to ensure the safe evacuation of embassy staff, as well as Afghans who worked as interpreters or in other support roles.

A Pentagon official said Tuesday that around 4,000 would soon be in place.

At the White House, National Security Advisor Jake Sullivan said the Taliban had pledged to allow "safe passage of civilians to the airport", adding: "We intend to hold them to that commitment."

Other governments -- including France, Germany, India and Australia -- have also organised charter flights.

But Washington has come under sharp criticism for its handling of the evacuations.

"The images of desperation at Kabul airport are shameful for the political West," German President Frank-Walter Steinmeier said.

US President Joe Biden and British Prime Minister Boris Johnson announced the Group of Seven would stage a summit next week on the crisis.

- 'The fear is there' -

The Taliban took effective control of the country Sunday when president Ashraf Ghani fled and the insurgents walked into Kabul with no opposition.

It capped a staggeringly fast rout of Afghanistan's major cities in just 10 days, achieved with relatively little bloodshed, following two decades of war that claimed hundreds of thousands of lives.

The collapse came as Biden moved to complete the withdrawal of US troops. He admitted Monday the Taliban advance had unfolded more quickly than expected but defended his decision to leave, and criticised Ghani's government.

US-led forces invaded the country following the September 11 attacks in 2001, in response to the Taliban giving sanctuary to Al-Qaeda, and toppled them.

This time around, the Taliban have sought to project an air of restraint and moderation.

"Those working in any part or department of the government should resume their duties with full satisfaction and continue their duties without any fear," they said Tuesday in a statement.

- Russia hails 'positive' meeting -

Russia's ambassador to Afghanistan Dmitry Zhirnov met with the Taliban in Kabul, hailing a "positive and constructive" meeting. 

European Union foreign policy chief Josep Borrell said the bloc would "have to talk" to the Taliban.

But Canadian Prime Minister Justin Trudeau said Ottawa would not recognise a Taliban government.