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Fannie and Freddie likely would have to borrow the money they use to pay Treasury, however, and Mr. Coburn said that would be at a higher interest rate than Treasury would pay — raising questions about whether that is a good deal for taxpayers.

Mr. Rutherford told reporters that he hopes the situation doesn’t come to that. He said he hopes Congress and the White House can reach an agreement on raising the limit before May 19.

That is unlikely, given that both sides have hardened their positions since 2011, when Mr. Obama and House Speaker John A. Boehner, Ohio Republican, worked out the deal that raised the debt ceiling more than $2 trillion, in exchange for spending limits and the budget sequesters that are beginning to bite.

Mr. Obama said he would no longer negotiate, and Mr. Boehner said any debt increase would have to be matched dollar-for-dollar with spending cuts.

Robert Bixby, executive director of the Concord Coalition, a nonpartisan deficit watchdog group, said he predicts some sort of deal, though nothing like the “grand bargain” both sides were flirting with in 2011.

“I think both sides want to avoid the ugliness of the summer of 2011,” he said, predicting “a mini-deal that might tweak sequestration and raise the debt limit.”

“I wouldn’t expect a big debt limit increase or much change in sequestration, but enough to save face on both areas. In other words, more of the same,” he said.