Greek banks "cannot afford" the current uncertainty, the chairman of one of Greece's biggest lenders said on Monday, warning that failure to reach an agreement with creditors would be "disastrous".

"The Greek banking system cannot afford, cannot continue the current situation of every day losing hundreds of millions of euros by depositors that are fearful of a Grexit," Eurobank's Nikolaos Karamouzis told the BBC in an interview.

"It's very difficult. Every day we have to try very hard to calm our staff, we have to try very hard to calm our clients, we have to address the issue that we have plenty of collateral to address the capital outflows, the deposit outflows," he said.

He said that Greek banks' dependence on funds from the European Central Bank to make up for the deposit outflow "is not a sustainable situation".

"To me, to the management of the bank, to the thousands of employees of the bank, we are absolutely convinced that Greece has to proceed to find an agreement with our creditors and European partners. For Greece, no agreement would be a disastrous event," he said.

Greece's central bank said Greek business and private bank deposits had fallen by nearly 30 billion euros ($34 billion) between December and April, to 128 billion euros.

Greek savers have been pulling their money from national banks at even higher rates in recent weeks, amid fears Athens may fail to reach agreement with its creditors to free up bailout funds needed to honour a June 30 debt repayment.