For months during the Democratic presidential nominating contest, Hillary Clinton has resisted calls from Senator Bernie Sanders to back a single-payer health system, arguing that the fight for government-run health care was a wrenching legislative battle that had already been lost.

But as she tries to clinch the nomination, Mrs. Clinton is moving to the left on health care and this week took a significant step in her opponent’s direction, suggesting she would like to give people the option to buy into Medicare.

“I’m also in favor of what’s called the public option, so that people can buy into Medicare at a certain age,” Mrs. Clinton said on Monday at a campaign event in Virginia.

Mr. Sanders calls his single-payer health care plan “Medicare for all.” What Mrs. Clinton proposed was a sort of Medicare for more.

The Medicare program covers Americans once they reach 65. Beneficiaries pay premiums to help cover the cost of their coverage, but the government pays the bulk of the bill. Mrs. Clinton’s suggestion was that perhaps younger Americans, “people 55 or 50 and up,” could voluntarily pay to join the program.

She made the remarks as she continues to face a determined challenge on the left from Mr. Sanders, forcing her to essentially fight a two-front war as she seeks to turn her attention to Donald J. Trump and the general election. While Mr. Sanders trails by a substantial number of delegates, his effect continues to be felt in the race as he pressures Mrs. Clinton to adopt more progressive positions.

“Bernie Sanders’s campaign is having an effect on Hillary Clinton’s policies,” said Steve McMahon, a Democratic political consultant from Purple Strategies. “From a progressive point of view, that’s exactly what was hoped for and that is exactly what is happening.”

The idea of allowing people to buy in to Medicare has been discussed in policy circles and in Congress for decades. Mrs. Clinton’s husband, former President Bill Clinton, floated a similar proposal in 1998, including it in his State of the Union address that year. The strategy has been embraced by many advocates of single-payer health care as a way to move more Americans into the existing government system. An incremental expansion of Medicare was the hoped-for strategy of Medicare’s original authors.

But it is a new idea in Mrs. Clinton’s presidential campaign. She has called for arange of health policy overhauls to preserve and expand the Affordable Care Act. She has proposed expanding financial protections for people with high health care costs and expanding subsidies to help middle-income people buy their own insurance. She also has proposed a package of policies to lower the price of prescription drugs.

But more recently, she has moved further. In February, she began discussing the possibility of a “public option,” a government-run insurance plan available to people shopping on the existing marketplaces. That idea was considered when the Affordable Care Act was being debated in Congress, but it was ultimately removed from the law.

Mrs. Clinton’s latest suggestion regarding Medicare, first reported by Bloomberg News, takes another step by proposing that Americans still in their prime working years be given the opportunity to obtain the same government insurance that is provided on a universal basis to their older peers.

“The politics in the short term are good,” said Jonathan Oberlander, a professor of health policy at the University of North Carolina at Chapel Hill. “Medicare is a popular program — people don’t know too much about health reform, but people know about Medicare.”

Before the Affordable Care Act, people older than 55 tended to have difficulty buying their own insurance, because insurance companies saw them as bigger risks. Since the health law passed, insurance plans have been banned from discriminating against people based on health history, but they can charge premiums that are three times as much as younger adults are charged.

Moving more older adults into the Medicare program could have the effect of lowering insurance costs for younger people, as Mrs. Clinton suggested. But the exact dynamics would depend on how the program was structured.

Mrs. Clinton did not say, for example, whether lower-income Americans choosing Medicare would receive help paying their premiums, as they do when they buy private plans on the new marketplaces in the Affordable Care Act. Without such subsidies, Medicare might be an affordable option only for wealthy or very sick customers. In 2008, the Congressional Budget Office estimated that a Medicare buy-in program for 62- to 64-year-olds would cost about $7,600 a person.

Medicare tends to cover a wider range of doctors and services than commercial plans, but it also has more holes — it has no limit on how much patients can be asked to spend out of their own pockets, for example. That means conventional Medicare would have different benefits and financial protections than private plans that are available to the 50-to-65 population.

“It’s not a bad concept to be playing with and thinking about,” said Linda J. Blumberg, a senior fellow at the Urban Institute, a left-leaning policy research group. “But there are some complexities that need to be ironed out.”

Taking a more progressive stance on health care could serve Mrs. Clinton in a general election matchup against Donald J. Trump, the presumptive Republican nominee. He has offered a range of views on the subject. While he says that he wants to repeal the Affordable Care Act, he also has called for a system that would ensure that all Americans have health coverage.

Mr. Sanders is pressing on in his campaign and hopes to influence the party’s platform at the July convention even if, as expected, he does not become the nominee. He has already pulled Mrs. Clinton to take more progressive stances on several issues, including supporting a higher minimum wage and opposing the Trans-Pacific Partnership trade deal and the Keystone XL pipeline.

Although Mr. Sanders viewed Mrs. Clinton’s latest health care proposal as progress, he said on Tuesday that it was not sufficient.

“Secretary Clinton’s proposal to let the American people buy into Medicare is a step in the right direction, but just like her support for a $12 minimum wage, it is not good enough,” Mr. Sanders said in a statement that described her idea as “Medicare for some.”