Congressional leaders across the political spectrum have pledged to approve emergency military support to Israel following Hamas’s surprise attack over the weekend, but protracted gridlock over federal spending and the Republican leadership crisis engulfing the House have raised questions about how and when such aid could be delivered.

In the four days since Hamas fighters broke through the barriers surrounding the Gaza Strip and blitzed nearby Israeli towns, killing civilians and seizing hostages, every Democratic and Republican leader on Capitol Hill — including the two G.O.P. candidates vying to become speaker of the House — has stated unequivocally that the United States should send Israel whatever supplies it needs to retaliate. More than 390 members of the House have signed on to a resolution laying out that position, reflecting the broad bipartisan consensus around backing the Jewish state.

President Biden made it clear on Tuesday that he would be asking Congress to take “urgent action” to help the security interests of allies and partners.

But there is no such unanimity regarding the amount of emergency assistance that ought to be approved by Congress, or whether it should be tethered to additional military assistance that Mr. Biden has requested to continue to support Ukraine, which is in limbo for the first time since Russia invaded that country.

Democrats and Republicans are divided, even within their own party ranks, over whether the broad political consensus behind backing Israel’s war effort might help break the logjam over sending more military supplies to Ukraine, or simply complicate the heavily politicized negotiations over doing so. A growing number of Republicans are resisting continued aid for Ukraine and suggesting that any additional assistance would have to come with major Democratic concessions, including more restrictive immigration laws.

There are also questions about whether the House — rudderless with no permanent speaker after Republicans ousted Representative Kevin McCarthy from the post last week — is prepared to act. G.O.P. lawmakers are set to meet on Wednesday to select a new nominee for speaker, but they are deeply divided, which could prolong the process of electing a new top official to preside in the House.

To date, leaders of both parties have operated under the assumption that Representative Patrick T. McHenry, the North Carolina Republican who was named temporary speaker after Mr. McCarthy’s removal, has only limited governing powers that do not include the right to take major legislative action. But there is no precedent for the current situation, and according to congressional experts, an acting speaker could do whatever a majority of the House agreed to do.

“The House as a legislative body is not powerless. It can come into session and consider legislation,” said Sarah Binder, a congressional expert and professor of political science at George Washington University. “At the end of the day, it’s a matter of interpretation.”

Leading members of both parties have used the war between Israel and Hamas to slam the far-right Republicans who voted along with all Democrats to remove Mr. McCarthy from the speakership, arguing that their tactics had neutered the House’s ability to respond to the worst fighting between Israelis and Palestinians in over half a century.

“It’s my hope that our Republican colleagues get their act together,” Representative Hakeem Jeffries, Democrat of New York and the minority leader, said on CNN on Sunday.

The Biden administration has been sending mixed messages about how imminently Israel might need emergency supplies.

On Monday night, National Security Council spokesman John Kirby told reporters that there were “no specific efforts to seek additional authorities or anything from Congress” underway regarding Israel.

But in a speech on Tuesday, Mr. Biden said that he would be going to Congress to ensure that his administration had adequate funding to keep Israel supplied with ammunition and the interceptors it utilizes as part of its Iron Dome missile defense system.

It is unclear what that request to Congress might look like, and whether it would involve aid to other countries. Israel currently receives about $3.3 billion in annual foreign military financing. Earlier this year, the House and Senate passed versions of the annual defense bill that directed $80 million to Iron Dome, among other allocations for the country’s missile defenses.

The uncertainty has prompted some lawmakers to argue for pairing military assistance to Israel with aid to Ukraine, after an effort to send more funds and weapons to Kyiv foundered in the House last month after a majority of Republicans there voted against it.

“I think there’s discussion about putting Israeli funding with Ukraine funding, maybe Taiwan funding and finally border security funding. To me that would be a good package,” Representative Michael McCaul, Republican of Texas and the chairman of the Foreign Affairs Committee, told reporters Monday.

Senator Lindsey Graham, Republican of South Carolina, who has been a leading figure in Senate negotiations to secure military assistance for Ukraine, made a similar point.

“We have a chance to reset the world here: Defeat Putin in the Ukraine, dismantle Hamas now and tell the Iranians, if you escalate any more attacks coming from Iran we’re coming after you, anything short of that’s going to lead to an invasion of Taiwan by China,” he said on Fox News.

A sprawling package that combines global military assistance with U.S. border security could put lawmakers in both parties in a politically difficult spot. Right-wing Republicans who have balked at continued aid to Ukraine might be reluctant to oppose a measure to help Israel, for which many of them have expressed unequivocal support.

And some left-wing Democrats are opposed to providing military aid to Israel. On Tuesday, most of them endorsed the bipartisan resolution from Mr. McCaul and Representative Gregory W. Meeks of New York, the ranking Democrat on the Foreign Affairs panel, which declares that the House “stands ready to assist Israel with emergency resupply and other security, diplomatic, and intelligence support.”

But some Democrats, including those who opposed a House resolution earlier this year pledging unfailing support for Israel, refrained, citing concerns with funneling more military equipment to the country as it lays siege to the Gaza Strip, cutting off supplies of food, fuel, water and electricity.

“Instead of continuing unconditional weapons sales and military aid to Israel, I urge the United States at long last to use its diplomatic might to push for peace,” Representative Ilhan Omar, Democrat of Minnesota, one of the members withholding support for the resolution, wrote on X, formerly Twitter.

Several Republicans opposed to Ukraine aid have angled for addressing the war in Israel separately.

“Aid to Israel needs to be focused on Israel,” Representative Jim Banks, Republican of Indiana, wrote on X. “Lumping support with Ukraine funding needlessly holds up aid to our most important ally in their hour of need.”