Greek Prime Minister Alexis Tsipras will meet the leaders of the 18 other eurozone nations at a Brussels summit.
Louka Katseli, chair of the National Bank of Greece, told the BBC that banks were not under immediate threat of running out of money.
But she said the situation was serious, and without a deal would become severe.
Her remarks came as Mr Tsipras presented a Greek proposal for a deal during a conference call with German Chancellor Angela Merkel, French President Francois Hollande and EU Commission President Jean-Claude Juncker.