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.

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