Long queues formed outside the bank’s head office in central Kabul’s Shah-e-Naw district, and in provincial cities, such as Mazar-i-Sharif in the north, the western city of Herat and Kandahar in the south.
Customers at Mazar- i-Sharif were having trouble withdrawing more than $1,000 each, local people told the Bloomberg wire service.
Things could get worse, warns correspondent Jonathan Boone.
"There are a huge number of small depositors who certainly do not read the international media and it'll take them a while to even hear that this bank is in trouble," he says.
"If there is a panic and everyone wants their money back, then there won't be enough money to pay back the depositors and the bank will collapse."
The Kabul bank is enormous by Afghan standards, Boone says.
"It has 1.3 billion dollars (one billion euros) in deposits. It does most of the banking services for all or almost all the soldiers, policemen and many civil servants in this country ... That money has been very poorly handled by then owners of the bank."
Central bank chief Abdul Qadir Fitrat on Wednesday denied the reports and insisted that the bank has no liquidity problems.
But his statement has failed to reassure depositors. Anonymous bank officials told the AFP news agency that executives met Karzai on Thursday.