Speaker Mike Johnson on Tuesday encountered stiff resistance from Republicans as he embarked on a complicated and politically perilous strategy to push legislation through the House to send aid to Israel and Ukraine — all while beating back a threat to his own job.

Mr. Johnson, who has agonized for months over whether and how to advance aid to Ukraine that many in his party bitterly oppose, has settled on a multipart plan that will require everything to go right for him this week to prevail.

It aims to bring together a complicated mix of bipartisan coalitions and allow different factions in the House to register their opposition to pieces of the aid package without sinking the entire thing. And it would ultimately mean cobbling together just enough support from Democrats and mainstream Republicans to pass the legislation amid resistance from hard-right Republicans to Ukraine funding and among left-wing Democrats to unfettered aid for Israel.

Mr. Johnson plans to advance a legislative package that roughly mirrors the $95 billion aid bill the Senate passed two months ago with aid to Ukraine, Israel, Taiwan and other American allies — but broken down into three separate pieces that would each be voted on individually. There would also be a fourth vote on a separate measure containing other policies popular among Republicans, including conditioning Ukraine aid as a loan.

The strategy has run into a flurry of opposition from members of his own party, including one Republican, Representative Thomas Massie of Kentucky, who on Tuesday announced that he would join a threatened bid to remove Mr. Johnson from the top post.

“We’re steering toward everything Chuck Schumer wants,” Mr. Massie said of the aid package, referring to the Senate majority leader in explaining his decision to reporters after the meeting.

Mr. Johnson said he chose the approach because “every member, Republican and Democrat, can vote their own district and their own conscience on this thing.” It would allow, for example, Republicans who support aid for Israel but abhor aid for Ukraine to register each position separately, instead of forcing them to support or reject a combined foreign aid bill.

“The will of the House is to address it in single subjects, in regular order, in a regular process with an amendment process,” he said on “Fox and Friends.”

If all four pieces passed the House, they would then be folded into a single bill for the Senate to take up, in an effort to ensure that senators could not cherry-pick pieces to approve or reject.

Mr. Schumer, Democrat of New York, said he was “reserving judgment” on the legislative package “until we see more about the substance of the proposal and the process by which the proposal will proceed.”

House Republican leadership aides were working on Tuesday to put together the bill text. But some members of Mr. Johnson’s own party were already balking.

Mr. Massie, a libertarian-leaning lawmaker who was a leader of the effort to oust John A. Boehner as speaker nearly a decade ago, endorsed the ouster effort against Mr. Johnson led by Representative Marjorie Taylor Greene of Georgia, making him the second Republican to do so. In a closed-door meeting on Tuesday morning, he stood and told Mr. Johnson that he should announce a resignation date and allow Republicans to choose a new speaker before he relinquished the top post.

Representative Chip Roy of Texas, an influential conservative, said he was unhappy that Mr. Johnson had not included any border security measures in the foreign aid package — as he previously had insisted he would — and opposed the idea of sending the separate measures to the Senate in one package.

“Don’t use Israel as a way to force Ukraine down the throats of the American people without having border security,” Mr. Roy said.

The speaker’s plan, he said, “is being sold as an open process, but it’s all structured to achieve a final omnibus result which is going to be effectively similar to the Senate bill.”

Despite conservative frustration with the plan, it seemed far from clear that other Republicans would join Ms. Greene’s effort to oust another speaker, after Kevin McCarthy was removed from the position last October. Mr. Massie’s announcement on Tuesday prompted open frustration from mainstream Republicans and even some conservatives, who said they did not want to go through another painful ouster.

“We don’t need that,” said Representative Jim Jordan, Republican of Ohio. “No way. No way. We don’t want that. We shouldn’t go through that again.”

And a group of mainstream conservatives rallied to Mr. Johnson’s side. In a remarkable joint statement, all three leaders of the national security panels in the House — Representatives Michael McCaul of Texas, chairman of the Foreign Affairs Committee; Mike Turner of Ohio, chairman of the Intelligence Committee; and Mike Rogers of Alabama, chairman of the Armed Services Committee — endorsed the plan. Representative Tom Cole of Oklahoma, the Appropriations chairman, also signed on.

“There is nothing our adversaries would love more than if Congress were to fail to pass critical national security aid,” they wrote. “Speaker Johnson has produced a plan that will boost U.S. national security interests in Europe, the Middle East and the Indo-Pacific. We don’t have time to spare when it comes to our national security. We need to pass this aid package this week.”

Mr. Johnson defended his decision on Tuesday and flatly ruled out resigning.

“I regard myself as a wartime speaker,” he said at a news conference at the Capitol minutes after the closed-door meeting. “In a literal sense, we are. I knew that when I took the gavel. I didn’t anticipate that this would be an easy path.”

Former President Donald J. Trump, who gave Mr. Johnson a boost in a joint appearance last week but has recently upended the speaker’s legislative agenda, was circumspect when asked on Tuesday about Mr. Johnson while making a campaign stop after proceedings in his New York criminal case.

“Well, we’ll see what happens with that,” Mr. Trump said. “I think he’s a very good person.”

Even getting the national security package to the House floor for a vote would require extraordinary measures. Given mounting Republican opposition and the party’s razor-thin majority, it appeared certain that Mr. Johnson would not be able to bring up the bill, which requires a floor vote, without Democratic support.

The minority party almost never votes for a rule advanced by the majority in the House. But Democrats previously helped pave the way for legislation to suspend the debt ceiling, averting the nation’s first-ever default, and have since signaled that they might be willing to come to Mr. Johnson’s aid on issues of critical importance.

At least one Democrat, Representative Jared Moskowitz of Florida, suggested on Tuesday that he would move to save Mr. Johnson if hard-right Republicans forced a vote to remove him, a move that Representative Hakeem Jeffries, Democrat of New York and the minority leader, has suggested many in his party would consider.

“Massie wants the world to burn, I won’t stand by and watch,” Mr. Moskowitz wrote on social media. “I have a bucket of water.”

Michael Gold contributed reporting from New York.