In particular, Portman called on Cordray to come out in favor of bringing the CFPB's budget under the control of congressional appropriators, and actually replacing the director position with that of a bipartisan commission. The Ohio senator told the former Ohio attorney general that the bureau is in need of "basic accountability reforms" to bring its "unaccountable structure" under control.
Obama installed Cordray as director after Republicans last blocked his nomination in 2011 via a controversial recess appointment. He recently re-nominated Cordray to the position for a full term, as his recess appointment is currently slated to expire at the end of the year.
However, new questions about the viability of Cordray's existing stint have come into question. One week ago, a federal appeals court ruled that three recess appointments the president made the same day to the National Labor Relations Board were unconstitutional, drawing fresh questions on the viability of Cordray's appointment.
Portman said the ruling "casts a shadow over the CFPB and your nomination."
But so far, the White House has held firm, saying there was no reason for Republicans to block Cordray's nomination a second time around, and that the ruling did not affect Cordray's tenure. The CFPB, and Cordray's nomination, are currently facing a separate legal challenge.
CFPB backers, including the White House and Senate Democrats, have rebutted claims the CFPB is unaccountable, and argue efforts to change it are just attempts to weaken it.
The other Republican lawmaker to not sign on to the new CFPB blockade was Sen. Bob Corker (R-Tenn.), who sits on the Senate Banking Committee. His spokeswoman said he did not sign the letter because he is exploring legislative options to achieve more accountability at the bureau.
A CFPB spokeswoman declined to comment.