When Speaker Mike Johnson opened the floor for questions at a closed-door luncheon fund-raiser in New Jersey last month, Jacquie Colgan asked how, in the face of vehement opposition within his own ranks, he planned to handle aid for Ukraine.

What followed was an impassioned monologue by Mr. Johnson in which he explained why continued American aid to Kyiv was, in his view, vital — a message starkly at odds with the hard-right views that have overtaken his party. He invoked his political roots as a Reagan Republican, denounced President Vladimir V. Putin of Russia as a “madman” and conceded the issue had forced him to walk a “delicate political tightrope.”

Reminded by Ms. Colgan, a member of the American Coalition for Ukraine, a nonprofit advocacy group, of the adage that the only thing necessary for the triumph of evil was for good people to do nothing, Mr. Johnson replied that he kept a copy of the quotation framed in his office.

“That’s not going to be us,” he assured her. “We’re going to do our job.”

The exchange reflects what Mr. Johnson has privately told donors, foreign leaders and fellow members of Congress in recent weeks, according to extensive notes Ms. Colgan took during the New Jersey event and interviews with several other people who have spoken with him.

While the speaker has remained noncommittal about any one option, he has repeatedly expressed a personal desire to send aid to Ukraine — something he has voted against repeatedly in the past — and now appears to be in search of the least politically damaging way to do it.

The challenge for Mr. Johnson is that any combination of aid measures he puts to a vote will likely infuriate the growing isolationist wing of his party, which considers the issue toxic. Representative Marjorie Taylor Greene, Republican of Georgia, who has repeatedly said she would call a snap vote to unseat the speaker if he allowed a vote for Ukraine aid before imposing restrictive immigration measures, filed a resolution on Friday calling for his removal, saying she wanted to send him “a warning.”

Even if Ms. Greene follows through on the threat, Mr. Johnson could still hold onto his job. Representative Hakeem Jeffries of New York, the minority leader, has said he believed “a reasonable number” of Democrats would vote to save the speaker were he to face a Republican mutiny for acting on the Senate-passed aid package, though on Friday Mr. Jeffries said that had been “an observation, not a declaration.”

In a lengthy statement on Friday after Ms. Greene had filed her resolution and the House departed Washington for its Easter recess, Mr. Johnson said that when lawmakers returned in two weeks, they would “take the necessary steps to address the supplemental funding request.”

“We have done important work discussing options with members,” he said, “and are preparing to complete our plan for action.”

Privately, Mr. Johnson has expressed an interest in linking Ukraine aid to a measure aimed at forcing the Biden administration to reverse its moratorium on liquid natural gas exports, according to three people familiar with his deliberations who were not authorized to discuss them. Mr. Johnson pressed the issue at a White House meeting last month with President Biden and congressional leaders, arguing that by prohibiting new exports of domestic energy, the administration was increasing reliance on Russian gas, effectively enriching Ukraine’s enemy.

In that meeting, according to a person familiar with the comments, Mr. Johnson raised the case of Calcasieu Pass 2, a proposed export terminal that would be situated along a shipping channel that connects the Gulf of Mexico to Lake Charles, La., and would dwarf the country’s existing export terminals. The Biden administration in January had paused a decision on whether to approve it.

He has puzzled over whether to put the aid to a vote on the House floor packaged with assistance for other U.S. allies, including Israel and Taiwan, or allow lawmakers to vote on them separately to register their support for each individual nation.

With many Republicans bent on blocking aid to Ukraine, any legislation carrying it would need to be considered using a special procedure that bypasses House rules and requires a two-thirds majority for passage, relying heavily on votes from Democrats. But a combined aid package for both Ukraine and Israel like the one that passed the Senate last month could be doomed by a coalition of right-wing Republicans opposing the money for Kyiv and left-wing Democrats opposing aid for Israel.

Mr. Johnson has pondered imposing new sanctions against Russia. And he has debated how the money should be structured — straight assistance versus a loan — and whether it should be exclusively for lethal aid, a type of assistance that is more widely supported by his conference, or also include nonmilitary assistance.

“There is a big distinction in the minds of a lot of people between lethal aid for Ukraine, and the humanitarian component,” Mr. Johnson said at a news conference at the Capitol last week.

Both he and Representative Michael McCaul, Republican of Texas and the chairman of the Foreign Affairs Committee, have publicly floated the idea of paying for some of the aid by selling off Russian sovereign assets that have been frozen using legislation called the REPO Act.

Mr. Johnson has faced mounting international pressure to allow a vote on aid to Ukraine, fielding almost weekly visits and calls from NATO allies and pro-Ukraine activists both at his offices in Washington and Louisiana. When Prime Minister Donald Tusk of Poland visited Washington earlier this month, he had a sharp public message for the speaker.

“This is not some political skirmish that only matters here in America,” Mr. Tusk told reporters. “The absence of this positive decision of Mr. Johnson will really cost thousands of lives there — children, women. He must be aware of his personal responsibility.”

Meeting privately with Mr. Johnson in his office in the Capitol, President Andrzej Duda of Poland appealed to the Louisiana Republican’s respect for President Ronald Reagan, whose portrait hung beside the speaker during the meeting. Mr. Duda quoted Mr. Reagan extensively and praised his willingness to call out good versus evil during the Cold War, according to a person familiar with the comments who requested anonymity to describe them.

Some skeptical Ukraine backers, both on and off Capitol Hill, have fretted that Mr. Johnson’s agreeable comments have simply reflected his penchant for telling people what they want to hear. Early in Mr. Johnson’s tenure as speaker, lawmakers noticed that he had a habit of leaving listeners from warring factions with the impression he agreed with each of them.

Yet at the fund-raiser in New Jersey last month, he was fairly candid about his calculations.

Mr. Johnson told the audience that he was “working to figure out the best route forward,” Ms. Colgan recalled, adding that he said that half of House Republicans wanted to move it together as a package with Israel and Taiwan, and the other half wanted to do it on its own.

At a separate fund-raiser in Binghamton for a congressman in New York’s Hudson Valley last month, Christina Zawerucha, the executive director of the Together for Ukraine Foundation, and Anatoliy Pradun, the group’s president, who was born and raised in Ukraine, approached the speaker to press him on holding a vote.

Mr. Pradun had hoped to appeal to Mr. Johnson’s faith by telling him of the strong evangelical Christian community in Ukraine. But realizing they had little time to make their case, Ms. Zawerucha and Mr. Pradun instead gave the speaker a pin with the Ukrainian and American flags, showed him their poster advertising an upcoming interfaith vigil for Ukraine and implored him to schedule a vote on aid to Kyiv.

“He didn’t turn us away,” Ms. Zawerucha said. “He pointed at our poster and said, ‘I will take care of this. I will take care of this.’”

When Ms. Zawerucha relayed the interaction to fellow activists after the luncheon, they asked what she thought he meant.

“And at this point, I don’t know,” she said. “It’s been over a month since Speaker Johnson said he would take care of this. And a vote for Ukraine still has not been allowed on the floor.”

Julian Barnes contributed reporting.