(CNN) Most Americans want a version of Medicare for everyone, which is good news for presidential candidate Sen. Kamala Harris and other Democrats who support "Medicare-for-all," a national health care proposal in which people would get their insurance from a single government plan.

But there is an enormous catch: Not all Medicare expansion plans are the same and there is more support for some than for others.

Harris spoke in Iowa on Monday about embracing Medicare-for-all, even though in most instances it means eliminating private insurance. When pressed by CNN's Jake Tapper on whether that means eliminating private insurance, the senator from California answered affirmatively, saying she would be OK with cutting insurers out of the mix.

That could end up being a sticking point for a lot of Americans.

While a majority of Americans strongly or somewhat favored Medicare-for-all (56%) when they were asked about it in a Kaiser Family Foundation poll from January, significantly more (74%) favored creating a national government-administered health plan similar to Medicare that allows people to keep the coverage they have if they prefer it.

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