An emergency spending bill to speed military assistance to Ukraine and Israel faces its next critical hurdle in a rare Sunday vote, as Republican senators continue to jockey for a chance to make politically divisive revisions to the legislation.

The $95 billion measure, which would also fund humanitarian assistance for civilians in conflict zones and address threats to the Indo-Pacific region, easily cleared a series of test votes last week with bipartisan support, even though G.O.P. senators had voted to kill a previous version of the bill that included measures to address border security.

Some of the Republican senators who have helped nudge the foreign aid bill along have warned that their support for its final passage — a vote leaders hope to hold on Tuesday — depends on having the ability to propose changes on the floor, including several related to the border. While they have acknowledged it is unlikely such proposals could secure the 60 votes needed to amend the bill, the exercise of voting on them still allows them to show the G.O.P. base — and former President Donald J. Trump, whose criticism doomed the earlier version — that they tried to force the changes through.

But by Saturday night, party lawmakers had yet to finalize a list of potential revisions that Democratic leaders would accept.

“We still hope our Republican colleagues can work with us to reach an agreement on a reasonable list of amendments,” Senator Chuck Schumer, Democrat of New York and the majority leader, said on the floor, adding: “We will keep working on this bill until the job is done.”