President Biden on Friday delivered to Congress a $105 billion request for military aid, mostly for Israel and Ukraine, essentially daring lawmakers who oppose parts of the proposal to vote against an overall package that he said will ensure “American security for generations.”

Mr. Biden is betting that Republicans who oppose more aid to Ukraine will not vote against legislation that also includes more than $14 billion to help Israel defend itself against terror attacks from Hamas. And he is hoping that progressive Democrats who do not want to support Israel’s military operations will not vote against additional aid for Ukraine.

Also included in the request are billions of dollars to bolster security along the U.S.-Mexico border, security aid for Taiwan and a fund for humanitarian assistance in hot spots around the world.

The legislative gamble is playing out against a global split screen: wars are raging in Europe and the Middle East, while in the United States, the House has been in a state of chaos for more than two weeks as Republicans struggle to select a speaker.

Mr. Biden’s effort to convince a fractured Congress to back his funding request will test the argument he made to the American people in a speech to the nation Thursday night, when he pressed for global engagement in a deeply unstable world. In that address, he said that the cost to the United States of refusing to invest in that engagement will be high.

“These conflicts can seem far away,” Jake Sullivan, the president’s national security adviser, said in a call with reporters on Friday, as the president’s request was sent to Congress. “But the outcome of these fights for democracy against terrorism and tyranny are vital to the safety and security of the American people.”

Administration officials expressed confidence that there is broad, bipartisan support in both chambers of Congress that should come together quickly to pass the president’s emergency spending proposal.

But the initial reaction on Capitol Hill included some angry words from Republicans, who accused the president of trying to force them into voting for a Ukraine war effort they don’t believe in.

“It’s ridiculous that they’re even trying to lump them together. It’s absolutely outrageous,” said Representative Anna Paulina Luna, Republican of Florida, one of many hard-right G.O.P. members who have urged swift passage of a bill to help arm Israel but have consistently voted against military assistance for Ukraine.

For months, right-wing Republicans have been arguing that programs to arm Ukrainian fighters are siphoning money away from other domestic security objectives and pitching the United States closer to a head-on confrontation with Russia. They have taken a very different tone, however, when it comes to arming Israel in its fight with Hamas, characterizing that as a matter vital to international security.

Even some House Republicans who have continued to support Ukraine assistance took issue with the White House’s approach, and warned that the G.O.P. leadership there would try to split up the package.

“The House should split it up,” Representative Darrell Issa, Republican of California and a regular backer of Ukraine assistance, said of the president’s proposal, adding: “To bring the two together is inherently divisive.”

Last month, more than half of Republicans in the House voted against a bill to replenish a $300 million account dedicated to training and equipping Ukrainian fighters, after several joined hard-liners like Ms. Luna to try to force the administration to be more forthcoming with a plan for victory. Some of those members said that Mr. Biden’s White House speech Thursday night had done little to address their concerns.

“What I want to see is a commander in chief that can sit in the Oval Office and tell America, ‘Here’s what we’re going to do, here’s what we want to spend that money on and here’s what we want our result to be,’ ” said Representative Brad Wenstrup, Republican of Ohio. He said he wanted to see the various security crises covered under Mr. Biden’s request for funds handled separately, in order to get those answers on aid to Ukraine.

If Congress approves Mr. Biden’s combined approach, Ukraine would receive $61.4 billion for military and economic assistance, while Israel would get about $14.3 billion to bolster its air and missile defenses, including the Iron Dome system, which has protected the country from incoming Hamas rockets.

The request also includes more than $9 billion for humanitarian assistance in Israel, Gaza and Ukraine and $7.4 billion for security to support Taiwan and other allies in the Indo-Pacific. It also includes almost $14 billion to fortify border security operations in the United States — a nod to Republican demands that domestic security be considered alongside global conflicts.

Shalanda Young, the director of the Office of Management and Budget, said some of the administration’s request for humanitarian aid would be used to help victims of the violence in Israel and Gaza.

“You’ve already seen a commitment from this administration in making sure humanitarian aid gets to those in Gaza,” she said. “That aid will continue robustly as Congress funds more humanitarian aid.”

Mr. Biden’s allies in the House leadership backed the president’s approach on Friday, predicting that the strategy would prove successful in the end.

“There’s support for it,” said Representative Adam Smith, Democrat of Washington and the ranking member of the Armed Services Committee.

The proposal has already drawn some opposition from the left wing of the Democratic Party, where more than a dozen members have called for a cease-fire in the Israel-Hamas war. They argue that billions in military assistance for Israel would simply accelerate a conflict they are trying to bring to an end.

“Sending more money for weapons doesn’t get us to peace,” Representative Delia Ramirez, Democrat of Illinois, said in an interview.

Ms. Ramirez sounded wary about the money marked for security along the U.S. border with Mexico, saying “more money for enforcement at the expense of immigrants is unacceptable to me.”

The proposal includes about $13.6 billion for additional border patrol agents, immigration judges and other border security measures.

The way the House handles the president’s request may depend in part on how — or whether — Republicans can bring an end to their impasse over choosing a new speaker.

Legislative activity has been at a standstill since a handful of hard-line Republicans orchestrated the ouster of Representative Kevin McCarthy, Republican of California, from the post two and a half weeks ago. No spending package, regardless of its scope, is likely to get a vote until either a replacement is named or members vote to give Representative Patrick McHenry, Republican of North Carolina and the current acting speaker, the power to put bills on the floor.

In the meantime, Senator Chuck Schumer of New York, the Democratic leader, has pledged that the Senate will vote swiftly on Mr. Biden’s full national security package. That could put pressure on the House to follow suit, though the Senate’s appropriations panel is not expected to begin working on legislation to fund the president’s request until the end of the month.

“The Biden administration’s request sends a clear message to America’s friends and allies that we have your back,” he said.

Senator Mitch McConnell of Kentucky, the top Republican in the Senate, also signaled his support for combining the foreign aid requests.

“This is all interconnected,” he told reporters earlier this week.

But even in the Senate, there is some opposition. Senator Tom Cotton, Republican of Arkansas and a frequent critic of the president, issued a blistering statement.

“President Biden’s slush fund proposal is dead on arrival, just like his budgets,” he said, claiming that Mr. Biden’s proposed funding for humanitarian needs in Gaza and elsewhere would help finance Hamas terrorists. “Senate Republicans will take the lead on crafting a funding bill that protects Americans and their interests.”