Armed police in pick-up trucks have been stationed outside the main branch of Kabul Bank as customers continue to withdraw money amid fears the Afghan bank may collapse.
Barbed wire has also been placed across the road to hold back the crowds.
The run on the bank began earlier this week after allegations of corruption and mismanagement, although officials have maintained the bank will not fail.
Meanwhile the US Treasury has denied it is preparing to bail out the bank.
"While we are providing technical assistance to the Afghan government, no American taxpayer funds will be used to support Kabul Bank," Deputy Treasury Secretary Neal Wolin said.
The BBC's Mark Dummett said that queues started forming outside the main branch before dawn.
Customers have been told that the branch is the only one that holds US dollars but they can only withdraw a maximum of $10,000.
"I am withdrawing my money because I am taking a trip and I also believe my money will not be safe in this bank," said 26-year-old Waheed Shirzoi.
The governor of the Central Bank, Abdul Qadir Fitrat, told the BBC the queues were only slightly bigger than normal. Most people wanted to withdraw their salaries or take out money to pay for presents before the Eid festival, he said.
Kabul Bank, Afghanistan's largest commercial bank, is reported to have run up huge debts that it cannot afford to pay.
Last week, newspaper reports said the bank's two top executives - chairman Sherkhan Farnood and chief executive Khalilullah Ferozi, who each own 28% of the bank's shares - had been replaced and Mr Farnood ordered to surrender $160m worth of property purchased in Dubai.
However, Mr Fitrat has said the men resigned voluntarily as new regulations did not permit shareholders to hold executive positions.