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.”