"The ECB clearly has the resources to preclude any further bank runs and would intervene if rates get out of control," said Bergstrand.
The ECB could provide additional liquidity to banks or buy more government bonds. The central bank is widely expected to lower interest rates at its next policy meeting in July.
However, ECB President Mario Draghi has said the central bank cannot make up for a lack of action by government leaders. He has called on EU leaders to strengthen the European financial system by forming a so-called banking union.
"The ECB will be loath to act before it is sure what political decisions will be taken, domestically and by the eurozone as a whole," said Andrew Milligan, head of global strategy at Standard Life Investments.
Banks in the United States have very little exposure to Greece, and experts say the exposure to Spain and Italy is manageable. But U.S. banks would not be immune if contagion spreads to banks in Germany and France.
"U.S. banks have indirect exposure, but in the context of a financial crisis, those indirect exposures become very important," said Bryson.