The decision by Republicans in Congress to torpedo a bipartisan border deal they demanded has left the fate of aid to Ukraine and Israel in peril, closing off what had been seen as the best remaining avenue on Capitol Hill for approval of critical military aid to American allies.

The political paralysis in the face of pleas from President Biden, lawmakers in both parties and leaders around the world for quick action raised immediate questions about whether Congress would be able to salvage the emergency aid package — and if so, how.

In the Republican-led House, where many conservatives have resisted new aid to Kyiv and the border deal negotiated in the Senate, lawmakers were set on Tuesday to vote on legislation sending $17.6 billion in military assistance to Israel. But that measure was facing steep resistance from hard-right Republicans who complained that the money was not paired with spending cuts, as well as Democrats and Mr. Biden, who has threatened a veto, calling the bill a cynical attempt to pre-empt the security legislation brokered in the Senate.

The $118.3 billion Senate bill, which is set for a test vote on Wednesday, also appeared to be dead even before it reached the floor, after an increasing number of Republicans — even those who led the charge to negotiate it — declared they would vote to block it.

“Joe Biden will never enforce any new law and refuses to use the tools he already has today to end this crisis,” Senator John Barrasso of Wyoming, the No. 3 Republican, said. “I cannot vote for this bill. Americans will turn to the upcoming election to end the border crisis.”

The grim reality appeared to be sinking in for the nation’s allies and diplomatic leaders who for months have anxiously watched and hoped for momentum on Capitol Hill for approving aid, despite clear signs that the strong bipartisan consensus for such a measure was fraying.