Tehran said on Sunday it was preparing to house Afghans escaping their country in three border provinces, but warned that they should return when it is safe to do so.
Hossein Qasemi, an Iranain interior ministry official, said: “Camps have been built in border areas in three provinces.
“We expect those Afghan refugees to return home when the situation improves in Afghanistan.”
The Associated Press reported that civilians fearing the return to hardline Islamist rule including the scrapping of women’s rights were rushing to leave Afghanistan.
Some were already heading for Pakistan with photos taken at the Friendship Gate crossing point in the Pakistani border town of Chaman showing groups and lorries waiting to cross.
Queues at cash machines were also reported with ordinary Afghans attempting to withdraw their savings.
However some ATMs stopped distributing cash as hundreds gathered in front of private banks.
Many of the poorest people who had left homes in the countryside for the presumed safety of the capital were stuck in open spaces, along roads, or in car parks throughout Kabul with only tents for shelter.
Hospitals have also been struggling to cope with the numbers of people wounded in the fighting, with 17,000 treated in July and the first week of August in facilities supported by the International Committee of the Red Cross, the aid agency said.
Before the sudden and unexpectedly rapid advance of the Taliban over the last week, Iran had been an attractive destination for Afghan citizens looking for work.
The oil-rich neighbour's promise of jobs and relative safety had appealed to many in a country ravaged by decades of civil war.
But Iran's economy is itself flagging, with US sanctions forcing Tehran to push many of the more than 2 million Afghan refugees in the country to return home.
Additional reporting by agencies
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