At a news conference on Capitol Hill, Bernie Sanders said Hillary Clinton’s college affordability plan could “revolutionize” higher education. (Andrew Harrer/Bloomberg News)

Hillary Clinton took a major step Wednesday toward soothing tensions with Bernie Sanders, embracing key elements of a college affordability plan that was a rallying cry for many backers of her vanquished Democratic primary rival.

Clinton’s plan stopped short of the free-college-for-all idea pushed by Sanders, but it came close — proposing to eliminate college tuition for students from many middle-class families who attend public colleges and universities, as part of a broader goal of making higher education debt-free for all Americans.

And her willingness to embrace it, even after she had laid out a more limited student debt agenda long ago and spent months criticizing Sanders’s plan as unworkable, was seen by his camp as a significant concession. Aides for both campaigns, which have been negotiating for weeks, said other policy shifts could follow, while a long-awaited endorsement of Clinton from Sanders could come as early as next week.

Sanders, in a rare moment of praise for a Clinton policy proposal, held a news conference at a Capitol Hill office used by his campaign to say that Clinton’s college affordability plan could “revolutionize” higher education.

His campaign manager, Jeff Weaver, called the Clinton plan “a tremendous step forward” that would serve as “a signal to the people who supported Bernie Sanders that they’re being listened to.”

The thaw comes as aides to the presumptive Democratic nominee work ahead of the party’s national convention to minimize tensions that have lingered in the wake of a bruising primary in which Sanders portrayed Clinton as the embodiment of a rigged political system. While polls show liberal Sanders supporters have largely rallied behind Clinton since she secured the nomination, Clinton aides still see winning Sanders’s blessing as critical in mobilizing voters on the left, particularly in closely contested battlegrounds.

Sanders has remained an official candidate for the White House despite Clinton having effectively clinched the nomination weeks ago. Aides argue that his status gives him leverage in pushing Clinton to the left on his policy priorities and on the Democratic platform — although patience with Sanders has begun to wear thin in some quarters.

He received some groans Wednesday during a closed-door appearance before the House Democratic Caucus on Capitol Hill, according to a person who was in the room.

Sanders has dialed back his criticism of Clinton considerably since the end of the primary season. But in recent television interviews, he has suggested the timing of an endorsement of Clinton has as much to do with her as him. Late last month on MSNBC, for example, he said Clinton needed to show the American people that she “is prepared to stand with them as they work longer hours for lower wages, as they cannot afford health care, as their kids can’t afford to go to college.”

Asked at his news conference Wednesday what other ideas he was seeing movement on from Clinton, Sanders said: “We are working with Secretary Clinton on some initiatives.”

Among those, according to aides, is health care. During the primaries, Sanders touted a universal, “Medicare-for-all” plan that Clinton is not expected to embrace — but there are steps short of that that could probably satisfy the Sanders camp.

Sanders can already claim some other victories in the Democratic platform, which a committee has been crafting in advance of the convention this month.

At the urging of Sanders’s representatives, provisions have been adopted to raise the minimum wage to $15 an hour, break up big banks and expand Social Security through increased taxes­ on high-income earners.

Sanders is also planning a big push at a Democratic Party platform meeting this weekend in Orlando to adopt an amendment opposing the Trans-Pacific Partnership, a trade deal being pushed by President Obama but disliked by many pro-Democratic unions. Presumptive Republican nominee Donald Trump has tried to make inroads with union voters by vowing to undo such free-trade deals.

Sanders said Wednesday that the policy changes­ in the Democratic platform were just as vital, and actionable, as the college tuition change.

“My job, and I think the job of many of us is to make sure the platform is not just a piece of paper — that it is a document that forms the foundation for Democratic Party proposals,” he said.

Sanders’s effectiveness in pressuring Clinton, and Clinton’s willingness to compromise with him, were apparent Wednesday in her unveiling of the new college affordability plan.

During the primaries, Clinton repeatedly said that she didn’t think taxpayers should foot the bill for children from wealthy families to attend college. She often singled out the children of Trump.

Clinton’s new plan would ensure that families with income below a certain level will pay no tuition at in-state public colleges and universities, according to her campaign. The plan ultimately would cover more than 8 in 10 families, the campaign said.

The expansion of the tuition plan would be introduced on a sliding scale: At the start, students from families making $85,000 a year or less could attend a four-year public college or university tuition-free. The income threshold would increase by $10,000 a year every year over the next four years, the campaign said, meaning that by 2021 all students with a family income of $125,000 or less could pay no tuition.

Before Wednesday, Clinton had proposed only that community colleges be tuition-free for all working families.

The original cost of Clinton’s college plan — which she said aims to help students “drowning in debt caused by ever-rising college costs” — was $350 billion over 10 years. A Clinton aide said the expansion would raise the cost by more than $100 billion. Any such plan would require congressional approval.

Clinton is also proposing a three-month moratorium on the repayment of federal student debt upon taking office. That would allow time to refinance or restructure high-interest debt.

Democratic members of the Senate Health, Education, Labor and Pensions Committee, which would handle any education reform bill, said they were warm to the Clinton proposal.

But they stopped short of saying that Sanders deserved all the credit for it. Sen. Robert P. Casey Jr. (D-Pa.) said that it would have been difficult “to find the dollars” for Sanders’s more expansive plan but that the idea was sound.

Sanders arguably did more to call attention to Clinton’s new plan on Wednesday than she did.

Besides holding a news conference to tout it, he put out a press release headlined: “A Revolutionary Step Forward for Higher Education.”

Clinton’s release, by contrast, carried the relatively milquetoast banner: “Hillary for America Unveils Further Measures to Make Debt-Free College Available to All.”