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.
France is the euro zone's second-largest economy, and has so far remained relatively untroubled by the recent debt crisis which has engulfed smaller economies like Greece.
The falls in French banking stocks have left many analysts scratching their heads as to the cause.
"The two potential explanations for the weakness in French banks yesterday – and especially SocGen – to our minds fail to justify the magnitude of the move," Matt King, analyst at Citi, wrote in a research note, which rated SocGen "buy", Thursday.
Some in the market are concerned about a potential increase in the size of the European Financial Stability Facility (EFSF), a multilateral institution created in response to the sovereign debt crisis. The EFSF will issue bonds backed by the euro zone's AAA-rated economies, and there are worries it will increase an already significant French debt burden.
"Even if France were to be downgraded – despite S&P and Moody’s denials, perhaps as a result of EFSF expansion – it would not create large capital losses or liquidity shortages for French banks," King insisted.
"On paper, even under haircut scenarios, sovereign debt exposure for banks looks painful yet manageable but market stress in itself can ultimately be self-fulfilling," warned the BoA analysts.
"The most plausible explanation is that the capital pressures at French insurer Groupama – whose high exposure to equities is well known – led to large selling in French bank equities," King argued.
"But even the 30 percent drop over the past month in a bank’s equity does not create immediate liquidity problems."