The White House's push for tax cuts made crucial progress on Tuesday as Senate Republicans rallied behind a budget proposal the party needs to pass to keep alive its hopes of enacting sharp reductions in tax rates later this year.

Senate Republican leaders earned a series of much-needed victories Tuesday, first with the return of ailing Sen. Thad Cochran (R - Miss.) and later with an announcement from Sen. John McCain (R - Ariz.) that he would back the budget resolution in order to help passage of tax cuts. Senate Republicans are now hopeful they can agree on a final budget resolution later this week, which is a key procedural step to help them pass a tax cut plan later in the year without relying on support from any Democrats.

"If we get the Republicans we need, which is virtually every single one of them...we will get that largest tax cut in the history of our country," President Trump said Tuesday in a speech to the Heritage Foundation. "And you will see things happen like have never happened before. We will have employment. We will have jobs. We will have companies moving back into our country."

Even with the breakthroughs on Tuesday, many hurdles remain. Republicans still haven't written a tax cut plan, they haven't identified trillions of dollars in tax deductions they plan to eliminate, and they haven't sorted out how to ensure that the majority of any tax cuts don't benefit primarily the wealthy. Still, none of those things would have mattered if they failed to pass a budget resolution.

"I support the Senate budget resolution because it provides a path forward on tax reform," McCain said in a statement. "I have long supported efforts to fix our burdensome tax system and hope Congress will produce meaningful reform that simplifies the tax code, strengthens America's middle class and boosts our economy."

A budget resolution lays out nonbinding spending priorities, but it also clears the way for the Senate to later approve $1.5 trillion in tax cuts over 10 years with just 50 votes. Republicans control just 52 votes in the 100-seat Senate, and they would normally need 60 votes to win passage of such sweeping changes. This is a crucial step in a closely divided Senate, where few if any Democrats are expected to support tax cuts.

Debate began on the measure midday Tuesday after a party-line vote to move the process forward.

If Republicans succeed in passing the budget, they will need to reconcile differences betwen their measure with a budget resolution that has already passed the House of Representatives.

Approving the budget would help shore up ties between Senate GOP leaders and President Trump, who is angry at Republicans' failure on health care and bent on Congress approving a tax-reform package by the end of the year.

Democrats are trying to stay unified to block the tax plan, but they will need GOP defections in order to have any success. Numerous Democrats have said the tax plan would mostly help the wealthiest Americans and they have disputed White House assertions that corporate tax cuts raise wages by thousands of dollars. They have also said the tax cuts would add trillions of dollars to the government's debt, eventually hurting economic growth because of increased borrowing costs.

Even though McCain's support is expected to help smooth passage of the budget resolution, he and Trump have been bickering for weeks and these tensions are likely to fester. McCain said late Tuesday he remains very concerned about what he believes are inadequate levels of military funding in the budget, but he said that can be addressed outside the process the Senate plans to vote on this week.

"It is an absolute requirement that we adequately fund the men and women who are serving in the military. I've said it 50 times, I'll say it again: Men and women in uniform are being killed and wounded because we have refused to fund them adequately in order to do their job," he said.

Asked whether he was close to a deal to satisfy his demand for more spending, McCain sounded optimistic.

"Yeah, sure!" he said. "We're having those conversations with leadership."

The GOP's success on the final budget vote is not yet sure, though McCain's "yes" and Cochran's presence gives the party a wider margin for error. Republicans control 52 of the Senate's 100 seats, meaning they can lose two votes from their own party and still pass the budget.

The vote midday Tuesday on the motion to proceed was the first big test for leaders. It passed 50 to 47, with three senators absent and all other Republicans voting "yes."

Even Sen. Rand Paul (R-Ky.), who opposes the budget in its current form, voted "yes" to begin debate after a conversation with Trump.

If the House and Senate pass matching budget resolutions, focus will then shift to the House Ways and Means Committee, where Republicans leaders are trying to craft language that would serve as the first version of the tax cut bill. Trump has said he wants to slash the corporate tax rate, simplify the tax code, and deliver a tax cut to middle income families. But many of the details of how this would work need to be resolved. Trump did not provide more details of the plan on Tuesday in his speech, but he promised the changes would eventually jumpstart economic growth and raise wages for all Americans.

"We will lift our people from welfare to work, from dependence to independence, and from poverty to total beautiful prosperity," he said.