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A threat that military pay could be halted also was an issue this year when Congress faced a possible government shutdown during a budget dispute. Such a scenario was averted when lawmakers agreed to a stopgap spending measure to fund the government for the rest of the fiscal year.

“Don’t allow our military men and women to dangle over a fire and think they won’t get paid,” said Rep. Michele Bachmann, a Minnesota Republican who also co-sponsored Mr. King’s bill. “I am unwilling to do that to any military personnel or their families. That’s a dangerous game to play.”

Rep. Louie Gohmert, Texas Republican and another co-sponsor of the bill, said the president’s warning that Social Security checks could be withheld was nothing more than a scare tactic.

“The fearmongering needs to stop,” Mr. Gohmert said.

Social Security Administration Chief Actuary Stephen Goss told the House Budget Committee on Wednesday that the Treasury Department — not his agency — has the final say on whether or not to withhold Social Security benefit checks.

The question of whether or not Treasury can prioritize debt payments is murky. The department has argued that it lacks the legal authority to allow some debts to take precedence over others, saying to do so would be “a failure by the U.S. to stand behind its commitments.”

But in 1985, the General Accounting Office, now called the Government Accountability Office, said “Treasury is free to liquidate obligations in any order it finds will best serve the interests of the United States.”

The nonpartisan Congressional Research Service says the positions aren’t necessarily contradictory, suggesting “they merely offer two different interpretations of Congress’s silence with respect to a prioritization system.”

The organization issued a report Tuesday that said if Congress refuses to raise the government’s borrowing limit, the White House cannot do so unilaterally under the Constitution’s 14th Amendment.

Some political observers have argued that the 14th Amendment means the government cannot suspend payments and that the president can act to keep funding government even without congressional authorization.

The White House has been dismissive of using the 14th Amendment as a way to solve the debt limit impasse.

Stephen Dinan contributed to this report.