By Sarah Ferris, Alison Main and Lauren Fox, CNN

(CNN) — Senate Republicans narrowly approved President Donald Trump’s giant tax and spending cuts package Tuesday after a days-long grind to secure the support of key holdouts, leaving one major step to send it to his desk.

The vote comes after weeks of bitter GOP infighting, with Trump himself forced to intervene to convince fellow Republicans to back his plan. But the hard work for Republicans isn’t yet done: The bill must still be approved by the House, which is expected to return to Washington on Wednesday.

GOP leaders are racing to try to get the bill to the White House by July 4. The multi-trillion-dollar bill would unlock tax cuts and funding boosts for national security, partly paid for by the biggest cut to the federal safety net in decades.

Senate Republicans hurtled toward a final vote on the bill Tuesday after more than 24 hours of painstaking negotiations over changes to the package to win critical GOP support.

The vote at times appeared to be in flux, even hours before the final vote. Asked earlier Tuesday if GOP leaders had a deal to move ahead, Senate Majority Leader John Thune told reporters Tuesday morning, “I believe we do.” He added: “I’m of Scandinavian heritage. Always a bit of a realist. So we’ll see what happens.”

Republican Sen. John Hoeven of North Dakota said Vice President JD Vance – who arrived on Capitol Hill earlier Tuesday morning – was brought in to cast tie-breaking votes on several final changes to the legislation, including the massive package of negotiated changes from Senate GOP leadership known as the “substitute” amendment.

“We’ll need him on the actual substitute bill,” Hoeven said of Vance.

The burst of movement from the Senate GOP came after a full 24 hours of intense negotiating between Thune, Vance and the GOP holdouts, including Sen. Lisa Murkowski of Alaska.

Vance had been the latest senior Republican to try to personally woo Murkowski, one of the party’s critical holdouts, to back the giant package of tax and spending cuts. GOP leaders have spent days intensely lobbying the Alaska centrist with a lineup of policy sweeteners catered specifically to her state.

On Tuesday, she suggested they finally reached a deal.

“It’s in the hands of the people that operate the copy machine,” Murkowski told reporters when asked whether the vote was in the hands of the Senate parliamentarian.

Earlier, the parliamentarian – the chamber rules referee – determined that a food stamps-related carveout meant to win over Murkowski could remain in the legislation without running afoul of the chamber’s strict budget rules, while ruling that a provision meant to change federal cost sharing for Medicaid to benefit states like Alaska and Hawaii was not compliant, according to a Democratic source familiar with the ruling.

Thune and his leadership team spent the weekend pushing ahead with Trump’s agenda, though they didn’t yet have the votes. Now, their chamber has been voting on amendments to Trump’s bill for a full day — an unprecedented session that has frustrated Republicans and Democrats alike.

And it’s not even the final step before Trump can sign the bill: The narrowly divided House will need to pass the Senate’s exact version of the bill, though dozens of their own members dislike the bill. House GOP leadership have been privately telegraphing to the Senate for weeks that they should have simply adopted the House version — rather than largely rewritten it.

Still, if the Senate passes its version Tuesday, the House is expected to vote Wednesday on the measure, according to a GOP leadership source familiar with the plans.

It’s a rapid turnaround for House lawmakers, who are currently scattered across the country for the holiday recess, but multiple GOP sources said they believed they could get it done in the House this week and meet the president’s end-of-week deadline.

Both Thune and House Speaker Mike Johnson have been working furiously to deliver Trump his first major legislative win this week, so the president can sign it in a special ceremony on the Fourth of July.

CNN’s Morgan Rimmer and Tami Luhby contributed to this report.