The stalemate on Capitol Hill over sending vital military aid to Ukraine is posing a critical test for Representative Hakeem Jeffries, the New York Democrat and minority leader, who may hold the only remaining key to overcoming Republican opposition.

With the hard right dug in against the measure and the Republican speaker, Mike Johnson, unwilling to bring it up, Mr. Jeffries and other top Democrats are considering a variety of long-shot maneuvers to force it to the House floor. Once there, the $95 billion foreign aid bill that commits $60.1 billion to Ukraine funding would almost certainly have enough bipartisan support to pass.

But that would require Mr. Jeffries to quell a backlash against the package in his own party, where left-wing lawmakers are deeply opposed to the $14.1 billion in security assistance it would provide to Israel. Their resistance could make it more difficult for Mr. Jeffries to orchestrate an end run around Mr. Johnson, which would likely require near-total Democratic unity, as well as some Republican support, to have any chance of success.

The legislation passed the Senate this month with solid bipartisan support, and enjoys similarly broad backing in the House, where a vast majority of Democrats are in favor and dozens of Republicans have consistently voted for bolstering Ukraine’s war effort.

But Mr. Johnson has rejected bipartisan appeals to bring the foreign aid bill to a vote, a move that at least one right-wing lawmaker has said would prompt her to call for his removal. The speaker and many other Republicans have said they have no intention of considering more assistance for Ukraine without first securing the United States border with Mexico through severe immigration restrictions. And with former President Donald J. Trump egging them on from the campaign trail, there is a potentially high political price to pay for any Republican who crosses them.

That has led Democrats and some Republicans to begin discussing the prospect of banding together to go around Mr. Johnson and force the measure to the floor. The options for doing so all require a majority, or 218 lawmakers, to go along. That threshold has become increasingly feasible for Democrats given the shrinking Republican majority; the G.O.P. now controls only 219 votes, while Democrats hold 212.

But with most Republicans exceedingly unlikely to cross their own party on the matter, Mr. Jeffries cannot afford many defections. That means he would have to persuade most if not all of the progressive skeptics in his party to stifle their concerns with aid to Israel long enough to help bring the measure to the floor.

“I want to support Ukraine, but it can’t be at the expense of more Palestinian children,” said Representative Delia Ramirez, Democrat of Illinois and one of the potential holdouts.

Though dozens of liberal Democrats harbor misgivings about sending military aid to Israel, they are split over whether merely to vote against the bill or to attempt to block it altogether.

“I would not be willing to sign onto a new discharge petition,” Representative Summer Lee, Democrat of Pennsylvania, said in an interview, referring to a maneuver that can force a bill to the floor if 218 lawmakers sign a petition demanding it. She said that financing military equipment for Israel as it launches a new assault on Rafah, Gaza’s southernmost city, “with no debate and no substantive change in behavior from us, and no conditions, I think, is not the right way of addressing this continued conflict.”

Yet others who expressed similar concerns said they could not let their misgivings about the Israeli offensive keep the House from voting on a viable measure to arm Ukraine.

“The only way to get the Ukraine money is to bring this whole package to the floor,” said Representative Jim McGovern of Massachusetts, the top Democrat on the Rules Committee. He said he might ultimately oppose the bill but believes “the process has to go forward.”

In interviews, members of the Congressional Progressive Caucus guessed that most left-wing Democrats would ultimately side with Mr. McGovern.

“I think you’ll find most are going to sign it,” Representative Raul Grijalva, Democrat of Arizona and a past chairman of the progressive group, said of a discharge petition.

Even some of the chamber’s most vocal critics of unconditional aid, such as Representative Alexandria Ocasio-Cortez, Democrat of New York, hinted that they would not stand in the way of the bill receiving an up-or-down vote on the floor.

“I think we’re all on the same page here, collaborating as a caucus,” she said in an interview.

Democratic leaders, eager to shore up every last vote they can, have not yet determined what mechanism they will use to try to force Mr. Johnson’s hand.

Last week, Mr. McGovern filed a measure that Democrats could use to try to bring the foreign aid bill to a vote through a discharge petition, possibly by late next month. Mr. Jeffries has also approached some progressive members to ask if they would keep their signatures on an older discharge petition that Democrats prepared last spring to address the debt limit, if that became the mechanism for getting the foreign aid bill to the floor.

The thinking, according to aides familiar with the private conversations who described them on the condition of anonymity, is that some left-wing Democrats might find it more awkward to remove their names from a petition they have already signed than to withhold their signatures from a new measure.

Yet Democratic leaders are also hedging their bets with Mr. McGovern’s legislation, in case a fresh petition helps them attract the support of mainstream Republicans, whose eventual buy-in is critical to making any gambit work.

At this point, no Republican has expressed interest in joining what, for them, would amount to a mutiny against their own party. Some top targets, like Representative Brian Fitzpatrick of Pennsylvania, are promoting alternative legislation to send military assistance to Ukraine and Israel with no humanitarian aid for civilians, in exchange for certain border provisions, including a one-year revival of the Trump administration’s Remain in Mexico policy.

Seasoned House Republicans are skeptical that will change anytime soon.

“I’ve only seen a discharge petition work once in the 25 years I’ve been here,” said Representative Mike Simpson, Republican of Idaho. “It’s just not something that Republicans are willing to run around the speaker on.”

Others suggested that the Democrats’ maneuvering could turn up the heat on Mr. Johnson enough to make him change his mind.

“Any speaker can stand in the way of the majority will on the House floor for a period of time — but not permanently,” Representative Patrick T. McHenry, Republican of North Carolina, said in an interview with CBS News this week. “My hope is that the speaker will come around to seeing this in a very sensible way.”