On Tuesday, the Senate Banking Committee’s ranking member, Mike Crapo, R-Idaho, said he thinks Cordray’s appointment was unconstitutional.

Crapo and fellow Republicans want to replace the bureau’s single directorship with a commission or board of trustees, subject the bureau to the congressional appropriations process and give other federal regulators more influence in the bureau’s rulemaking and policy decisions.

“The Dodd-Frank Act specifically elevated the director of the (bureau) so that he or she holds unique power to determine the agency’s budget and mission priorities without any public debate or input from Congress,” Crapo said.

He said moving to a board format would bring more transparency to the agency, and that including it in the federal appropriations process would let Congress know how its money is being spent.

Republican Sen. Mike Johanns of Nebraska asked Cordray whether he would be willing to appear before members of the powerful Senate Appropriations Committee “to walk through your budget document and just answer questions about that document and spending habits of your agency.”

Cordray answered that he has testified before banking and finance committees in the House and Senate dozens of times in the past two years, but if senators want him to appear before another committee, “we could consider that.”

He added, “If there are measures you suggest, including walking through the budget and being subject to more transparency and accountability that would be satisfactory to you and your colleagues, we could certainly consider those things.”

Cordray shared Tuesday’s nomination hearing with Mary Jo White, Obama’s choice to lead the U.S. Securities and Exchange Commission. White, a former U.S. attorney in Manhattan who made her name prosecuting accused terrorists and mobsters, promised “unrelenting” enforcement of Wall Street.

White assured senators that her private practice work for clients such as JPMorgan Chase and executives from Bank of America and Goldman Sachs over the past decade would not present a conflict of interest.

“If I’m confirmed, the American public will be my client and I will work as zealously as I can,” she said.

White’s path to confirmation is expected to be much smoother than Cordray’s.