Advocates for private-school choice celebrated this week as a federal schools voucher bill moved closer to becoming law, a major milestone that eluded their movement during President Trump’s first term.

The House Republican budget proposal that advanced on Monday would devote $5 billion to federal vouchers for private-school tuition, home-schooling materials and for-profit virtual learning.

The program in the budget bill could bring vouchers to all 50 states for the first time, including Democratic-leaning ones that have long rejected the idea.

Supporters hailed the proposal as “historic” and a “huge win,” but some cautioned that there was still much legislative haggling ahead.

“Ultimately, every child, especially from lower-income families, should have access to the school of their choice, and this legislation is the only way to make that happen,” said Tommy Schultz, chief executive of the American Federation for Children, a private-school choice advocacy group.

Opponents of the proposal were stunned at its sweeping implications. While it is in line with President Trump’s agenda, it had been considered somewhat of a long shot to make it out of the House Ways and Means Committee, because of its cost.

The program is structured as a $5 billion tax credit, allowing donors to reduce their tax bill by $1 for every $1 they give to nonprofits that grant scholarships — up to 10 percent of the donor’s income.

The option to donate is expected to be popular with wealthy taxpayers.

The resulting scholarships could be worth $5,000 per child, reaching one million students. Any family who earns less than 300 percent of their area’s median income — which equals over $300,000 in some parts of the country — could use the funds, meaning a vast majority of families would be eligible.

The proposal could pass through the budget reconciliation process, and could become law with only 51 votes in a Senate where Republicans hold 53 seats.

In the wake of the Covid-19 pandemic, many Republican-led states passed new private-school choice laws, overcoming decades of resistance from teachers’ unions, Democrats and rural conservatives. Opponents have long argued that vouchers hurt traditional public schools, by decreasing enrollment and funding levels. And they have pointed out that lower-income neighborhoods and rural areas often have few private schools, making it difficult for many families to use vouchers.

“We are against giving people tax breaks to defund public schools,” said Randi Weingarten, president of the American Federation of Teachers, the nation’s second-largest education union.

She pointed out that while Mr. Trump and congressional Republicans have said they want to invest in work force education, artificial intelligence education and other priorities for student learning, they have consistently proposed cutting funding to public schools, which educate nearly 90 percent of American students.

“They don’t believe in public schooling,” she said. “What you’re seeing here is the fragmentation of American education.”

A boom in new private-education options, like virtual learning and microschools, has already changed the landscape — as has an influx of campaign spending from conservative donors, like the financier Jeff Yass, intended to build support for private-school choice.

Last month, Texas became the last major Republican-led state to pass such legislation. Advocates quickly shifted their focus to Congress and the opportunity to push a federal voucher bill.

Senator Bill Cassidy, Republican of Louisiana, is the sponsor of a Senate bill similar to the House proposal, and celebrated its inclusion in the budget package.

“Expanding President Trump’s tax cuts is about preserving the American dream,” he said in a written statement. “Giving parents the ability to choose the best education for their child makes the dream possible.”

But the proposal will still have to overcome opposition, on both the left and the right.

Advocates for public schools have said that the new generation of vouchers and education savings accounts, which are often available to relatively affluent families, are a subsidy to parents who can already afford private education.

In Florida, which has more children using vouchers than any other state in the nation, some public-school districts have experienced enrollment declines and are considering shutting down schools or cutting teaching positions.

Even some conservative parental-rights activists oppose the creation of a federal program, which they worry could create a regulatory pathway that could eventually be used to impose government requirements on home-schooling parents or private schools — for example, by requiring standardized testing, which is not mentioned in the current proposal.

“The federal government should extricate itself from K-12 education to the fullest extent possible,” said Christopher Rufo, a leading crusader against diversity programs in schools, and a supporter of school choice. “It’s best left to the states.”

Harvard University struck a respectful but firm tone in a letter to the Trump administration on Monday, arguing that the university and the administration shared the same goals, though they differed in their approaches. It was latest move in an extraordinary back-and-forth between the school and the federal government in recent weeks.

The letter from Alan M. Garber, Harvard’s president, was sent a week after the Trump administration said it would stop giving Harvard any research grants.

Last month, the university took the government to court over what it has called unlawful intrusion into its operations. But on Monday, Dr. Garber’s tone was softer, saying he agreed with some of the Trump administration’s concerns about higher education, but that Harvard’s efforts to combat bigotry and foster an environment for free expression had been hurt by the government’s actions.

Dr. Garber said he embraced the goals of curbing antisemitism on campus; fostering more intellectual diversity, including welcoming conservative voices; and curtailing the use of race in admissions decisions.

Those goals “are undermined and threatened by the federal government’s overreach into the constitutional freedoms of private universities and its continuing disregard of Harvard’s compliance with the law,” Dr. Garber said in the letter to Linda McMahon, the secretary of education.

The university’s response came one week after Ms. McMahon wrote to Harvard to advise the university against applying for future grants, “since none will be provided.” That letter provoked new worries inside Harvard about the long-term consequences of its clash with the Trump administration.

“At its best, a university should fulfill the highest ideals of our nation, and enlighten the thousands of hopeful students who walk through its magnificent gates,” Ms. McMahon wrote. “But Harvard has betrayed its ideal.”

Rolling through a roster of conservative complaints about the school, Ms. McMahon fumed about the university’s “bloated bureaucracy,” its admissions policies, its international students, its embrace of some Democrats and even its mathematics curriculum.

Ms. McMahon referred to Harvard as “a publicly funded institution,” even though Harvard is private and the vast majority of its revenue does not come from the government. She suggested that the university rely more on its own funds, noting that Harvard’s endowment, valued at more than $53 billion, would give it a “head start.” (Much of Harvard’s endowment is tied up in restricted funds and cannot be repurposed at will.)

“Today’s letter,” Ms. McMahon wrote, “marks the end of new grants for the university.”

In Dr. Garber’s letter on Monday, he said that the university had created a strategy to combat antisemitism and other bigotry, and had invested in the academic study of Judaism and related fields. But he said the university would not “surrender its core, legally-protected principles out of fear of unfounded retaliation by the federal government.”

He denied Ms. McMahon’s assertion that Harvard was political.

“It is neither Republican nor Democratic,” he said of the university. “It is not an arm of any other political party or movement. Nor will it ever be. Harvard is a place to bring people of all backgrounds together to learn in an inclusive environment where ideas flourish regardless of whether they are deemed ‘conservative,’ ‘liberal,’ or something else.”

Although Harvard is the nation’s wealthiest university by far, officials there have warned that federal cuts could have devastating consequences on the campus and beyond. During Harvard’s 2024 fiscal year, the university received about $687 million from the federal government for research, a sum that accounted for about 11 percent of the university’s revenue.

The government can block the flow of federal money through a process called debarment. But the procedure is laborious, and the outcome may be appealed. Experts on government contracting said Ms. McMahon’s letter indicated that the administration had not followed the ordinary procedure to blacklist a recipient of federal funds.

Harvard officials are aware that, even if they challenge the administration’s tactics successfully in court, Mr. Trump’s government could still take other steps to choke off money that would be harder to fight.

The federal government often sets priorities for research that shape agencies’ day-to-day decisions about how and where federal dollars are spent. Some academics worry that the government might pivot away from fields of study in which Harvard has deep expertise, effectively shutting out the university’s researchers. Or the administration could simply assert that Harvard’s proposals were incompatible with the government’s needs.

Jessica Tillipman, an expert on government contracting law at George Washington University, said that it can be difficult to show that the government is using a back door to blacklist a grant recipient.

“You basically have to demonstrate and point to concrete evidence, not just a feeling,” she said.

Still, she said, Ms. McMahon’s letter could offer Harvard an opening to contest a protracted run of grant denials.

“It’s not as hard to prove,” Ms. Tillipman said, “when you have a giant letter that said, by the way, we aren’t giving you these things anymore.”