Senior Biden administration officials privately believe only weeks remain before a lack of additional Ukraine funding starts to become a serious battlefield concern -- a scenario they are trying to avoid with public warnings and a major speech from President Joe Biden himself.

The race for the House speakership set off by the historic ouster of Kevin McCarthy on Tuesday portends potentially serious consequences for Biden's efforts to secure Ukraine funding, leaving the administration looking for solutions.

Publicly, officials say they remain convinced the majority of Americans -- including in Congress -- support sustained assistance for Ukraine. Yet the maneuvering this week demonstrates the persistent concern that American assistance to Kyiv could soon slow. Biden on Wednesday hinted that administration officials have been searching for workaround methods of providing Ukraine assistance should the White House's funding requests go unmet.

"It does worry me," Biden said when asked Wednesday whether he was concerned about delivering Ukraine the aid he's promised. "But I know there are a majority of members of the House and Senate in both parties who have said that they support funding Ukraine."

Administration officials have been warning Congress it must urgently approve additional funds to aid Ukraine's war efforts -- "obviously time is of the essence," stressed one official.

Yet without even the chance for a vote on a new speaker until at least next week -- and no clear pathway for a vote on new Ukraine assistance after that -- the prospects of a new assistance package in the near-term appear slim.

Privately, officials believe a weekslong period where Congress were to hypothetically operate without a permanent House speaker -- and not be able to legislate -- would not be hugely concerning as it pertains to Ukraine funding.

Much more troubling, they said, would be if lawmakers begin to approach the end of the length of the most recently passed continuing resolution -- which runs out November 17 -- without any realistic prospects of approving additional funding for Ukraine.

Feeling the urgency, Biden told reporters Wednesday he was planning an address laying out the imperative of continued support for Ukraine.

"I'm going to make the argument that it's overwhelmingly in the interests of the United States of America that Ukraine succeed," he said. White House officials provided no other details about the speech, including when Biden might deliver it.

He also told reporters, "There is another means by which we may be able to find funding" should Congress fail to approve new aid, suggesting an effort to find workarounds as lawmakers stall on approving the assistance. Neither the president nor his aides would elaborate on the alternative means. Pentagon officials reiterated on Wednesday that the most effective mechanism for supporting Ukraine is still through the Presidential Drawdown Authority, which needs to be replenished with the supplemental funds that are now in limbo.

For now, White House officials are closely monitoring the speaker's race unfolding on Capitol Hill for signs of how the Ukraine debate might play out. While publicly staying mum on any specific lawmakers eying the job, aides have been tracking closely the dividing lines among Republicans over providing more assistance.

The leading contenders for the job have voiced different positions on Ukraine. A pro-Kyiv group that grades Republican lawmakers on their support for Ukraine has assigned a B grade to Rep. Steve Scalise of Louisiana, who has voted for previous assistance packages. It assigned Ohio Rep. Jim Jordan an F, the lowest grade, pointing to his previous votes against Ukraine funding.

Jordan, who announced his bid for speaker on Wednesday, said he opposed a new aid package for Ukraine.

"I'm against that," he told CNN's Manu Raju when asked if he would bring a Ukraine package to the floor if elected speaker.

For months, top administration officials have voiced confidence that new Ukraine funding would eventually be approved by the Republican House, despite the protestations of several conservatives. Jake Sullivan, the national security adviser, has predicted a process of "to-ing and fro-ing" among Republicans before eventually a "strong bipartisan support to continue funding Ukraine" emerges.

This summer, Biden requested an additional $24 billion in funding for Ukraine on top of the hundreds of billions already approved in US assistance. The stopgap measure that passed last weekend included no new Ukraine funds.

As lawmakers prepare for another government funding battle, there are growing calls in both chambers and from both parties for the White House to send Congress a single supplemental funding bill for Ukraine that would take American funding for the war in Ukraine through 2024 and past the election, two senior Republican congressional aides told CNN.

The bill wanted, the aides said, would be in the tens of billions of dollars -- far higher than the $24 billion requested by the White House in August. That argument has been made both privately to the White House, one of the aides said, but also publicly.

On Sunday Republican Sen. Lindsey Graham told CBS News: "I'm worried about next year, we will produce in the United States Senate, Ukraine funding $60 (billion) or $70 billion, not 24, to get them through next year."

In conversations with their European counterparts, American officials have sought to reassure them they are actively working with Congress on passing new Ukraine aid, according to people familiar with the matter.

One fear among administration officials is that an extended delay in providing new funding for Ukraine could lead to other nations backing off their own support, given the similar political pressures felt by other world leaders.

If there are continuing doubts about US funding and it slows, there will be cracks among the European supporters of Ukraine, a European ambassador in Washington told CNN, warning the anti-Ukraine forces in Europe will "rear their heads."

This instability is fueling the Russian perception that they can wait the West out, the ambassador added, and that divisions will form. The attitude is: "you are weak, if we test you, you will blink."

In a joint phone call with several NATO allies on Tuesday, Biden sought to reassure them of sustained American commitment to Ukraine.

"(I) made the case that I knew that the majority of the American people still supported Ukraine and the majority of the members of the Congress, both Democrat and Republican, support it," Biden said on Wednesday.

Still, the assurances previously provided to Biden by McCarthy about providing Ukraine aid have been scrambled by his departure from GOP leadership.

"Whoever the new speaker is, I'm sure it will have a significant impact on what additional support for Ukraine looks like. And we'll have to figure out how to pull together a coalition to pass something over there," Sen. John Thune, the No. 2 Republican in the Senate, told reporters.

Sen. Richard Blumenthal, a Connecticut Democrat, expressed similar concerns about Ukraine aid.

"I'm deeply concerned. There is an urgency, and we need to move forward whatever it takes. And border security is certainly on the table. We need to move forward with a real sense that time is not on our side. There's an urgency," he told CNN.

The uncertainty has rattled Ukrainian officials who have long feared a slowdown in international support.

Asked if she's afraid US military support will end, Ukraine's ambassador to Washington told CNN: "Of course, it worries us even if it's delayed."

"We [are] 100% dependent on our friends and allies," Ambassador Oksana Markarova said. "All we need, as Winston Churchill once said, are the tools. Give us the tools and we will finish the job."

Another Ukrainian official expressed confidence it will eventually work out.

"I'm not panicking, I still believe it's going to be fixed," the official told CNN, noting the Pentagon said Tuesday there is still over $5 billion worth of military equipment to disburse to Ukraine from the Presidential Drawdown Authority.

"We believe in US democracy," the official said.