For months, Republicans have insisted that any additional aid to the two US allies must also address the high numbers of migrants arriving at the US-Mexico border – a top voter concern.
But many Republicans promptly rejected the package when it was released on Sunday, even though it contained many of their priorities.
Former president Donald Trump, who has pressed them to reject any compromise, has made calls for tight controls of immigration a feature of his campaign to defeat Democratic President Joe Biden in the November election.
US homeland security chief survives House impeachment vote
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Only four of the Senate’s 49 Republicans voted for the bill.
“Some have been very clear with me they have political differences with the bill,” said Republican Senator James Lankford, one of the negotiators.
“They say it’s the wrong time to solve the problem, let the presidential election solve the problem.”
Independent Senator Kyrsten Sinema, another one of the deal’s authors, said she was baffled by the sudden shift in fortune.
“Three weeks ago, everyone wanted to solve the border crisis,” she said. “Yesterday, nobody did.”
Still, the defeat of the bill left open the possibility that Congress could yet provide much-needed aid to US allies. The Senate was expected to vote on Thursday on a US$96 billion package that strips out the immigration provisions but leaves the foreign aid intact.
An aide to Republican Senator Roger Wicker had predicted that a foreign-aid package would get well over 60 votes in the 100-seat chamber – a rare show of cross-party support.
But Senate leaders delayed action for hours instead, as Republicans sought to agree behind closed doors on a set of amendments that could revive the border debate and modify provisions to aid Ukraine and Israel.
Republican Senator John Cornyn said party members were aiming to come up with amendments on border security and on the distribution of aid to US allies.
“The question is what other amendments do people need in order to allow us to go to final passage,” the Texas Republican told reporters.
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Cornyn said he was optimistic about reaching a deal but expressed scepticism that an agreement could be ironed out later on Wednesday.
On Wednesday evening, the Senate Democratic Majority Leader Chuck Schumer announced that the vote would be put off at least until Thursday, to allow time for Republicans to determine their party’s way forward.
“We will have this vote tomorrow,” Schumer said.
Even if it passes, the aid faces uncertain prospects in the House of Representatives, as Republicans who control that chamber have balked at further support for Ukraine.
“We’ll see what the Senate does. We’re allowing the process to play out,” House Speaker Mike Johnson told reporters.
Johnson had said the border package would be “dead on arrival” in his chamber.
Johnson, meanwhile, said on Wednesday he would hold another vote to impeach Biden’s top border official, Homeland Security Secretary Alejandro Mayorkas, after a first attempt failed in a 214-216 vote on Tuesday.
“It was a mess what happened here, but we’re cleaning it up,” he told reporters on Wednesday.