The banks are in a vulnerable position because of outflows of billions of euros from deposits over the past six months.
Some 40 billion euros (USD 44 billion) has been withdrawn from Greek banks since December, according to the country's banks association.
And according to news reports today, the top four lenders -- National Bank, Piraeus Bank, Alpha Bank and Eurobank -- will undergo an asset quality review later this month.
Stress tests will follow in the autumn to determine the recapitalisation requirements of each bank with European rescue funds.
As of 2016, bank shareholders and depositors will foot the lion's share of recapitalisation costs -- a process known as "bail-in" -- instead of European taxpayers.
Greek banks were also recapitalised in 2013 with funds from the country's last EU-IMF rescue package.
The reopening of the market comes after senior EU and IMF auditors held their first meetings with Greek ministers to finalise the new three-year bailout which could be worth up to 86 billion euros (USD 94 billion).
The last session was on June 26, a few hours before Prime Minister Alexis Tsipras announced a referendum on the bailout conditions demanded by Greece's international creditors.
Over that weekend, Greeks rushed to bank cash machines, prompting the government to impose capital controls from June 29, followed by the closure of the banks and the stock exchange.
The aim was to protect the banking sector following the huge withdrawals by people nervous about Greece's economic and financial future.
For the duration of the crisis, Greece's banks have been relying on one lifeline: an ECB credit facility called an Emergency Liquidity Assistance (ELA).
The Frankfurt-based lender last week left the ELA unchanged at 90.4 billion euros
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