On Monday, President Obama nominated former Ohio Attorney General Richard Cordray to run the CFPB but Senate Republicans have said they won't confirm any nominee unless the Obama administration agrees to structural changes to the bureau.

The bill faces tougher prospects in the Democratic-controlled Senate. There, Banking Committee Chairman Tim Johnson (D-S.D.) has said he opposes changing the structure of the CFPB. However, in light of the Republican threats, there may well be additional negotiations among members of the Senate and between Congress and the White House.

The White House also has said the president would veto the bill as it emerged from the House.

CUNA and NAFCU both support allowing the bureau's decisions to be overturned by a majority of the Financial Stability Oversight Council, which is made up of the heads of all the financial regulatory agencies, including the NCUA.

But CUNA didn't take a position on whether the agency should be run by one person or by a five-member board.

When the House first passed the financial overhaul bill, it mandated that the CFPB be run by a five-member board but when the Senate passed its version it was changed to have a sand-alone director and the Senate version prevailed during the conference committee to reconcile the measures.

During the discussions of the financial overhaul bill, NAFCU opposed creating the CFPB with authority over credit unions and proposed that the bureau only regulate non-depository unregulated providers of financial services.

CUNA expressed concerns with the CFPB but didn't oppose it outright.

All credit unions have to comply with the rules issued by the CFPB, but the NCUA handles enforcement for institutions that have assets of $10 billion or less.

CUNA and NAFCU unsuccessfully sought to have the threshold raised to $50 billion, indexed for inflation. The trade associations had been successful in getting the threshold raised from its original figure of $1.5 billion.