FRANKFURT—The European Central Bank is struggling to limit the fallout from Greece's debt crisis on European financial institutions as it confronts a fresh problem closer to home: its own balance sheet.

Greek and other European banks likely have posted billions of euros in Greek government bonds and other securities as collateral for ECB loans. The central bank, which has largely replaced the private market as a source of funding for Greece, could face losses on its loan portfolio if Greek financial institutions fail and Athens...