Greek banks are currently reliant on a loan program offered by the European Central Bank known as Emergency Liquidity Assistance (ELA). This is designed to provide credit to financial institutions in the euro zone that are solvent but face "temporary liquidity problems."
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Last week, the ECB increased its ceiling for ELA to Greek banks by 2.3 billion euros ($2.6 billion) to 83 billion euros, in light of Greek savers continuing to pull out their savings.
However, the ECB is at liberty to restrict banks' access to the ELA if operations "interfere with the objectives and tasks of the Eurosystem." This decision would be taken by the ECB's Governing Council and would require a majority of two-thirds of the votes cast.
ECB President Mario Draghi and his colleagues have insisted that liquidity will be provided to Greek banks as long as they remain solvent. But if Greece fails to repay the 1.6 billion euros it owes to the International Monetary Fund at the end of this month, this could be in jeopardy.
In 2013 Cyprus turned to emergency liquidity assistance after the ECB refused to accept the country's sovereign bonds as collateral. Cyprus too faced warnings that the ECB's Governing Council was prepared to reject the continuation of ELA. The country later entered a rescue program and imposed capital controls to restore solvency to its banks.
—CNBC's Zack Guzman contributed to this report.