Even as Sen. Rand Paul was advancing plans to force a vote on declaring war on the Islamic State, House Speaker John A. Boehner said Thursday that it’s President Obama’s job, not Congress’s responsibility, to write a war authorization.

The conflicting signals suggest that despite the efforts of Mr. Paul, Congress will adjourn this year without voting on a new authorization for the use of military force (AUMF), or rules governing the war on terrorism — even though Mr. Obama and top congressional leaders have all said it’s critical to get it done.

Mr. Boehner said the debate should take place next year, and needs to begin at the White House.

“Historically, the commander in chief has identified the needs … written a new AUMF, sent it to the Capitol, and worked to build bipartisan support,” Mr. Boehner told reporters.

Mr. Paul, though, is still pushing for a vote this year, and he and his allies are determined to write and debate their own war declaration if Mr. Obama won’t.

On Thursday, the Kentucky Republican unexpectedly tried to force the fight as part of a debate on a water bill in the Senate Foreign Relations Committee.

The Kentucky Republican said he ultimately decided not to introduce the amendment, instead winning a commitment from Democrats, who control the Senate through the end of the year, to hold a hearing Monday and then vote on a new authorization later next week.

“I hope people are somewhat shamed, as well as the president should be shamed, into doing the right thing,” Mr. Paul said.

Lawmakers have been calling for a hearing for months, but they said Secretary of State John F. Kerry has been reluctant to appear on Capitol Hill and testify on the mission against the Islamic State. Even Sen. Robert Menendez, New Jersey Democrat and chair of the Senate Foreign Relations Committee, has said before that it’s difficult to work on a bill when the administration won’t provide witnesses.

It’s expected that Mr. Kerry will appear before the committee on Monday, but Sen. Tim Kaine, Virginia Democrat and a strong proponent of the need for congressional authorization of the current mission in Syria and Iraq, said the committee will move forward even if Mr. Kerry doesn’t attend.

“We’ll have a hearing with the best administration witnesses we can on Monday, whether or not they send over an AUMF draft,” he said.

The administration cites authority under previous authorizations from 2001 and 2002 to conduct strikes against the Islamic State in Iraq and Syria, but some lawmakers have said those are outdated and need to be updated.

Mr. Kaine, pushing back against Mr. Boehner’s timetable of next year, said action is needed now.

“If the House wants to allow the president to engage in unilateral war and blithely go home for the holidays when service men and women are dying in this war, that’s their choice,” Mr. Kaine said. “That’s not what we think is us carrying out the responsibility Congress is supposed to undertake.”

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