“Democrats are overwhelmingly for something that sounds like an all-government or significantly government-run system,” said Robert Blendon, a health policy professor at Harvard who studies public opinion. “At this stage, most people that are saying that don’t know what the ‘it’ is.”

The Republican slogan “repeal and replace” had some popularity, too, particularly with the Republican base. The United States health care system is expensive and full of dysfunctions, making it susceptible to anti-establishment critiques. But when Republicans took control of the government this year, they found themselves unable to coalesce around a single health care plan. Various options, which government analysts said would have resulted in at least 20 million fewer Americans with health insurance, were deeply unpopular with the public.

“Repeal and replace went from a very popular idea, getting rid of Obamacare, to a real proposal with winners and losers and trade-offs,” said Drew Altman, the president of the Kaiser Family Foundation, a health policy research group. Mr. Altman said that Democrats were benefiting from that shift now, as the defenders of the status quo. A splashy new health care proposal, with its own winners and losers and trade-offs, could reverse that political momentum, he said. “If they immediately switch the focus to their own sweeping health care plan, they could lose that opportunity.”

Still, the recent momentum suggests that single-payer may become a standard position for a certain type of liberal candidate. Joe Trippi, a Democratic campaign strategist, compared the issue to earlier opposition to the Iraq war. “It’s not proven yet that you could take the country there, but I think Democrats would respond more to a nominee who wants to take the country there than one who is half-loafing it,” he said.

For now, the top Democrats in Congress have yet to embrace a single-payer bill. Nancy Pelosi, the House minority leader, has emphasized the need to shore up the Affordable Care Act. Chuck Schumer, the Senate minority leader, said that single-payer was one of many options under study.

Their hesitance may reflect the challenging politics of the issue, despite the recent increase in support. The polls suggest that many voters may be more taken with the single-payer slogan than with the nuts and bolts. A recent survey by Kaiser found that initial support of 55 percent for single-payer dropped by about a third when supporters were told of criticisms that it might increase their taxes, give the government “too much control” over health care or eliminate the Affordable Care Act. Each of those critiques would probably be made prominently by Republican opponents of the policy.