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After weeks of criticizing Bernie Sanders’s plan for a single-payer “Medicare for all” health care system as budget busting and unrealistic, Hillary Clinton over the past few days has a new message for voters: Universal health care was her idea first.

“Now, before it was called Obamacare, it was called Hillarycare, as some of you might remember,” Mrs. Clinton said at a town-hall-style event in Clinton, Iowa, on Saturday, the second time in recent days she has used the line.

“We share the same goal, universal health care for every single American,” Mrs. Clinton said of her rival Mr. Sanders. “But we have a real difference about how to get there.”

Health care has become a major point of contention between Mrs. Clinton and Mr. Sanders, who are fighting to come out ahead in the Feb. 1 caucuses. Mr. Sanders’s plan would require over a trillion dollars a year paid for with a broad set of new taxes. His campaign says the plan would save middle-class families money on co-pays and deductibles despite the tax increase. Mrs. Clinton has hit Mr. Sanders for wanting to scrap President Obama’s signature domestic achievement and raise taxes on the middle class. She has promised to improve on the Affordable Care Act.

The strategy of reminding voters that she was at the forefront of pushing for universal health care could help Mrs. Clinton appear more authentic as she tries to compete with a populist liberal candidate who identifies as a Democratic socialist. But the new line in her stump speech could also evoke a tumultuous period during President Bill Clinton’s administration when the first lady was tasked with overhauling the health care system — and ultimately failed. Her proposals at the time aimed to provide universal health care, though they would have made employers responsible for their workers’ insurance, while Mr. Sanders envisions a government-run system.