The Senate on Sunday pushed a $95 billion emergency aid bill for Ukraine and Israel past a critical hurdle, with a bipartisan vote that kept it on track for passage within days.

The vote was 67-27 to move forward on the package, which would dedicate $60.1 billion to helping Kyiv in its war against Russian aggression, send $14.1 billion to Israel for its war against Hamas and fund almost $10 billion in humanitarian assistance for civilians in conflict zones, while addressing threats to the Indo-Pacific region. In a rare Sunday session, 18 Republicans joined Democrats to advance the measure, which leaders hope the Senate will approve as early as Tuesday.

“It’s no exaggeration to say the eyes of the world are on the United States Senate,” Senator Mitch McConnell, Republican of Kentucky and the minority leader, said on the floor on Sunday, appealing to his colleagues to back the bill. He maintained that U.S. allies “don’t have the luxury of pretending that the world’s most dangerous aggressors are someone else’s problem and neither do we.”

Senator Chuck Schumer, Democrat of New York and the majority leader, said on the floor, “We’re going to keep working on this bill until the job is done.” He commended Republicans who had backed the measure for “working in good faith to get this done” and asserted that it was “essential” for the Senate to pass the legislation. It had been decades, Mr. Schumer added, since Congress considered a bill “that so significantly impacts not just our national security, not just the security of our allies, but the very security of Western democracy and our ideals.”

But steep hurdles still remain for the bill in the Republican-led House, where it faces staunch opposition fueled by the “America First” stance of former President Donald J. Trump.

The bipartisan endorsement in the Senate came over the bitter opposition of right-wing Republicans who have railed against the measure, contending that the United States should not be continuing to send tens of billions of dollars to bolster Ukraine’s security, particularly without first doing more to secure its own border with Mexico against an influx of migration. They have continued to make the argument even after voting last week to kill a version of the aid bill that included a border crackdown, saying it did not go far enough.

Many Republican opponents have also taken issue with the billions of dollars the bill would devote to humanitarian aid for civilians in conflict zones, as well as $7.9 billion in economic aid to prop up Ukraine’s domestic infrastructure during wartime.

“We did spend four months promising the American people that we would secure our own border before we focused on other countries’ borders,” Senator Tom Cotton, Republican of Arkansas, said on “Fox News Sunday,” adding that he also had “serious concerns about the $19 billion of nondefense aid in there.” (Mr. Cotton voted last week with most of his party to kill the bipartisan border plan.)

Mr. Trump has stoked the resistance, urging Republican lawmakers to reject the bipartisan border plan, and egging on House G.O.P. leaders, who promised that it would be dead on arrival in their chamber. Mr. Trump has also made no secret of his opposition to funding Ukraine’s military campaign to push back a Russian invasion, a stance that he underscored during a campaign rally on Saturday by suggesting that, if re-elected, he would not defend U.S. allies against threats from Moscow.

Mr. Trump described the United States’ role in preserving the global democratic order as strictly transactional, declaring that if a NATO member failed to commit the requisite funds to bolster the organization’s collective security, he would refuse to defend them against a Russian attack. Of Russia, he added: “I would encourage them to do whatever the hell they want.”

NATO members are expected to commit at least 2 percent of their gross domestic product to military spending, a threshold that most of its member nations have not met.

Sunday’s action amounted to a repudiation of Mr. Trump’s stance by Democrats and a determined bloc of Republicans, led by Mr. McConnell, who have maintained that it is imperative that the United States continue to come to Ukraine’s aid militarily to send a signal to the rest of the world’s dictators.

“Deterrence is not divisible; American credibility is not divisible,” Senator Dan Sullivan, Republican of Alaska, said on the floor on Friday, adding, “You can’t say, ‘We’re going to be real strong in the Taiwan Strait, but you know — no problem in Ukraine.’ ”

Democrats predicted that enough Republicans would ultimately reject Mr. Trump’s pressure for the Senate to pass the measure.

“It has been hard to get Republican votes to support Ukraine, made very difficult by Donald Trump’s opposition to Ukraine funding, but I think we’re going to get this done,” Senator Christopher S. Murphy, Democrat of Connecticut, said on Sunday on CBS’s “Face the Nation,” warning that the United States would be “on the precipice of a disaster” that could lead President Vladimir V. Putin of Russia to threaten NATO allies if the Senate failed to pass the bill.

The bipartisan coalition that has carried the bill thus far will have to stick together for a few more votes before the Senate votes on approving the foreign assistance package and sending it to the House, where Speaker Mike Johnson is facing threats from the right to try to oust him if he puts a Ukraine aid bill on the floor.

The pushback from the hard-liners in the House G.O.P. is one reason Republicans have been so insistent on being allowed to propose revisions to the measure before voting on whether to pass it. The exercise of holding votes on partisan proposals, even if they are doomed to fail, is important, some Senate Republicans explained, to signal to the G.O.P. base where the party stands — and how impossible it would be to get all their demands through a Democratic-led Senate.

Before the Senate voted to advance the bill on Sunday, all but four Senate Republicans voted on a measure that would have stymied its progress, to protesting the fact that senators had not had more opportunity to propose changes to the bill. It did not pass.

If leaders can strike a deal on which measures to consider, the Senate could be in session for the long haul on Sunday voting on them. Aides were making plans over the weekend to ensure that senators would be able to watch the Super Bowl, carting extra televisions into the Capitol and ordering pizza, in case they were called to continue voting throughout the evening.

The Republican wish list of revisions to the foreign aid bill primarily focuses on the southwestern border. It includes a measure mirroring a restrictive border enforcement bill that the House passed last spring with Republican votes alone.

Democrats have responded with their own demands for revisions, such as a proposal from Senator Richard J. Durbin of Illinois, the Senate’s No. 2 Democrat, to give lawful permanent residence to certain undocumented immigrants brought to the United States as children.

After the demise of the bipartisan immigration plan, neither side is likely to be able to muster the 60 votes necessary to add such provisions to the final bill.

Among the other changes that Republicans have been demanding is a measure to strip the economic assistance for Ukraine from the bill. A subset of Democrats has also been angling for votes to limit the impact of Israel’s military campaign in Gaza, including a measure to prohibit Palestinian civilians from being forcibly displaced.