“Medicare for all” is the hottest idea in the Democratic presidential race for overhauling the nation’s health care system, and it is a phrase quite likely to be heard repeatedly at the first debates this week. But despite all the buzz, it turns out that the concept is dividing the 2020 field.

A new survey of the Democratic candidates by The New York Times finds that many of them prefer less sweeping changes than the Medicare for All Act, the single-payer bill introduced by Senator Bernie Sanders and supported by Senator Elizabeth Warren and several other presidential hopefuls. A majority of candidates in the survey — including former Vice President Joseph R. Biden Jr., the leader in early polls — said they would rather add a “public option” in the health care system that would compete with private plans.

The public option was once considered too far-reaching — champions of the idea in Congress could not muster quite enough support to include it in the Affordable Care Act in 2010. But it is now seen as a more moderate alternative to Mr. Sanders’s plan, which would all but eliminate private health insurance and enroll everyone in a government-run program.

Even some candidates who back the single-payer approach of Medicare for All told The Times they would also be willing to support a choice between some kind of public option and private insurance , an acknowledgment that many Americans like their private coverage and want the choice to keep it.