House Republicans considered a new stopgap funding proposal on Sunday aimed at averting a government shutdown at the end of the month, but it was unlikely that the plan, which would slash spending for most federal agencies and resurrect tough Trump-era border initiatives, could break the deep impasse on Capitol Hill.

The legislation presented to rank and file lawmakers in a conference call on Sunday night was the latest effort by House Republican leaders to find a way out of a daunting funding logjam that left their plans to consider annual spending bills in chaos last week and has put Congress on a path to a government closure on Oct. 1.

Leaders of both chambers concede that a stopgap measure will be needed to keep government agencies open after the Sept. 30 end of the fiscal year, when funding is set to lapse, because none of the 12 annual appropriations bills have yet cleared Congress.

The House proposal emerged from talks between right-wing and more mainstream Republicans and was meant to represent a compromise that both factions could embrace, avoiding a politically treacherous shutdown while also providing some funding cuts and border controls demanded by the ultraconservative House Freedom Caucus. But its fate was in serious doubt as reservations quickly emerged among some Republicans, and it was all but certain to be dead on arrival in the Democratic-controlled Senate even if the House could manage to pass it.

Republicans tentatively set a vote for Thursday, allowing time to overcome resistance in their ranks.

The proposal came after Speaker Kevin McCarthy said earlier Sunday that he intended to resurrect a Pentagon spending measure that stalled last week in an embarrassing setback for the speaker and try to push it to the House floor despite pledges by members of the Freedom Caucus to oppose the move unless their sweeping demands on spending were met.

His decision, announced on Fox News, was a bid to pressure far-right members to drop their insistence on steeper spending cuts or risk political heat for blocking the Pentagon funding bill. Pushing ahead with the military spending bill along with the new stopgap plan would be a major test of the ultraconservative Republicans, who are using the threat of a government shutdown at the end of the month to press their spending goals, as well as of Mr. McCarthy’s ability to unify the House G.O.P. with his job on the line.

“We’ll bring it to the floor, win or lose, and show the American public who’s for the Department of Defense, who’s for the military, who’s for giving them a pay raise,” Mr. McCarthy said on “Sunday Morning Futures.” He added that “any time a Republican wants to hold back and stop the floor from working when Republicans have the majority, that puts us in a weaker position to win in the end of the day.”

The proposed stopgap bill would extend funding through Oct. 31 and impose a nearly 8 percent spending cut on most federal agencies while exempting the Pentagon, veterans programs and disaster relief, resulting in a roughly 1 percent cut overall. It would include most elements of a tough immigration measure approved by the House in May, except for the E-Verify employment verification system, a plan that has drawn fire on several fronts.

It does not include additional assistance to Ukraine or added disaster aid, both of which are being sought by senators of both parties.

In the conference call, Mr. McCarthy told House Republicans that they could not prevail in a fight with the Senate if they could not pass a bill themselves, according to lawmakers who attended and described the discussion on the condition of anonymity.

The problem for Mr. McCarthy and his leadership team is that some of the most conservative House Republicans have said they do not intend to vote for a stopgap bill, known as a continuing resolution, under any circumstances, and he has just a handful of votes to spare.