Societe Generale's share price suffered a fall of around 8 percent Thursday morning after Wednesday's falls.
The French bank's market value plunged by around 15 percent Wednesday amid worries about its exposure to struggling economies elsewhere in the euro zone.
Thursday's fall followed a downgrade of European banks by Bank of America Merrill Lynch analysts.
"As long as EU peripheral debt issues remain in the headlines despite the best efforts of the ECB, banks will likely remain a focal point for negative risk appetite and we believe this will weigh against optically cheap valuations and low investor positioning," the analysts wrote in a research note, which downgraded the sector to neutral.
French banks BNP Paribas and Credit Agricole also continued their share price falls Thursday.
Chief executive Frederic Oudea told CNBC Wednesday that SocGen, France's second-largest bank, has only "limited exposure" to Spanish and Italian banks and is on target to meet Basel III regulations on capital provision. He added that he is confident in the French government's handling of the euro zone situation.
His words do not appear to have reassured the markets.
The insistence by Standard & Poor's and Moody's that France is not under threat of a downgrade has also failed to calm traders.