The recently published CFPB rule prohibits financial institutions from using arbitration agreements to keep consumers from being part of a class action suit. Immediately after its release, congressional Republicans announced their intentions to try to nullify the rule.

Republicans have been fighting the rule on two fronts. Noreika, an attorney who formerly represented banks, attacked the rule and left open the possibility that he would challenge it before FSOC. He said he was worried that the rule would affect the safety and soundness of the banking system.

He and CFPB Director Richard Cordray exchanged strident letters on the rule.

However, Noreika said Monday that he would have had to file a petition challenging the rule within 10 days of its publication in the Federal Register. The CFPB published the rule on July 10.

Nonetheless, Noreika said he remains concerned about the rule and urged Congress to take action.

"Ultimately, the rule may have unintended consequences for banking customers in the form of decreased availability of products and services, increased related costs, fewer options to remedy consumer concerns, and delayed resolution of consumer issues," he said. "The rule may turn out to be the proverbial straw on the camel's back."

Meanwhile, the House has passed a resolution that, if passed in the Senate and signed by the president, would nullify the rule under the Congressional Review Act.

The Senate is unlikely to consider it until September.

However, the state attorneys general urged the Senate to reject the resolution.

"The Arbitration Rule appropriately prevents consumer financial services companies from requiring their customers to agree to a contract that waives their right to join a class action filed against the company," they wrote in a letter.  "Businesses have used such waivers to force customers to resolve their disputes on an individual basis through secret arbitration proceedings conducted by private companies."

Credit union trade groups have said that while most credit unions do not include arbitration clauses in contracts, they favor rejection of the CFPB rule.

And the Trump Administration has signaled that if the Senate approves the resolution, the president's advisors would urge that he sign it.

"This legislation would protect consumer choices by eliminating a costly and burdensome regulation and reining in the bureaucracy and inadvisable regulatory actions of the CFPB," the Trump Administration said in a statement of administration policy.