The FDIC ordinarily insures deposits up to a cap of $250,000 per depositor, which leaves balances in excess of that being vulnerable in the event of a bank failure.

Anne Balcer, senior executive vice president and chief of government relations and public policy at the Independent Community Bankers of America (ICBA), told FOX Business that regulators should take a "prudent approach to any changes in FDIC insurance."

"It may make sense for Congress to look at the insurance limit cap and revisit raising it based on metrics demonstrating increasing deposit balances since the previous increase, but the tone coming from Treasury of picking winners and losers defies logic and is largely inappropriate," Balcer added.

The Mid-Size Bank Coalition of America (MBCA) sent a letter to the Treasury Department, FDIC, the comptroller of the currency, and the Federal Reserve to urge regulators to lift the deposit insurance cap for all institutions for two years, according to a report by Bloomberg.

Treasury Secretary Janet Yellen said in congressional testimony last week that uninsured deposits will only be backstopped by the government "if a majority of the FDIC board, a supermajority of the Fed board, and I in consultation with the president, determine that the failure to protect uninsured depositors would create systemic risk and significant economic and financial consequences."

Sen. James Lankford, R-Okla., noted that could make smaller banks a less attractive destination for depositors with funds above the $250,000 threshold.