(CNN) The emerging 2020 presidential campaign has deepened the Democratic party's divide over how the government should cover all Americans, further exposing the differences between those who want a total overhaul of the health care system and those who prefer a more incremental approach to fix it.

Sen. Kamala Harris of California renewed the party's debate over single-payer health care in a CNN town hall Monday, saying she's willing to end private insurance, which more than 170 million Americans use

"We need to have Medicare-for-all," said Harris. "Who of us has not had that situation, where you've got to wait for approval and the doctor says well, 'I don't know if your insurance company is going to cover this.' Let's eliminate all of that. Let's move on."

The momentum for such a proposal has grown among the left since Vermont Sen. Bernie Sanders' 2016 presidential campaign for the Democratic presidential nomination and Republicans' attempts to repeal the Affordable Care Act. Harris -- along with several other Democrats considered top-tier presidential candidates, including Sens. Elizabeth Warren of Massachusetts, Kirsten Gillibrand of New York and Cory Booker of New Jersey —have joined Sanders in supporting a bill to create a single, government-run insurance program modeled on Medicare.

On Tuesday, Rep. Pramila Jayapal, Democrat of Washington, said her goal is to introduce a similar House bill the second week of February.

Harris' comments appeared to put her firmly in favor of the most dramatic proposal to provide health care to everyone, rather than other options she's supported like allowing Americans to buy-into Medicaid.

But they again exposed the division within the party, as several Democratic senators -- from the rank-and-file members to leadership to other potential presidential contenders -- did not follow Harris' call to eliminate private health insurance.

"It would take a mighty transition to move from where we are to that," said Sen. Dick Durbin, the chamber's No. 2 Democratic senator in leadership.

"What most of us said we would support is a Medicare type plan -- a not-for-profit public plan that is available for everyone," said Durbin of Illinois. "I think that's a good first step."

Sen. Tim Kaine, the 2016 Democratic vice presidential nominee, claimed that "about 80%" of those who get insurance through their employers like their private insurance plan.

"I'm not going to say you have to give it up," Kaine said. "I think the idea is to offer a nonprofit insurance plan as an option."

Of eliminating private health insurance, Sen. Dianne Feinstein, Harris' senior Democratic colleague from California, said, "Well I'm not there."

Sen. Gary Peters, Democrat of Michigan, said he believes lawmakers should expand the ability for people to buy into a version of Medicare, adding he's a co-sponsor to a bill by Kaine and Sen. Michael Bennet of Colorado that would do just that. Sen. Chris Murphy, Democrat of Connecticut, plugged another bill that would expand access to Medicare, calling it "a lot more politically palatable and ultimately more popular than statutory prohibition private plans."

"It's a process -- you can't just pull the rug out from underneath everybody's feet," Peters said. "If you tell the American people you must do this, my experience has been a lot of people will say, 'You can't tell me to do anything.'"

Two likely Democratic presidential candidates declined to weigh in on the matter Tuesday. Warren told CNN that she "hadn't seen" Harris' remarks and said she would have to review them first. Booker, rushing to catch a Senate train, said when asked about Harris calling for an elimination of private health plans: "I'm not going to comment on that."

Many Americans don't want to give up their private plans for universal coverage. A recent Kaiser Family Foundation poll found that 74% of respondents favor creating a national health insurance program similar to Medicare, but allowing people to keep the coverage they currently have. Only 56% said they favor Medicare-for-all, in which all Americans would get their insurance from a single government plan.

But the question -- and answer -- is important to "Medicare-for-all" supporters on the left, who -- despite being pleased at the policy's growing popularity -- worry that it could be watered down in a crowded primary field.

When pressed by CNN's Jake Tapper on Monday if that means eliminating private insurance, Harris answered affirmatively, saying she would be OK with cutting insurers out of the mix. She also accused them of thinking only of their bottom lines and of burdening Americans with paperwork and approval processes.

"The idea is everyone gets access to medical care," she said.

On Tuesday, Sanders, an independent senator who caucuses with Democrats and is considering another run for the presidency, told CNN that he was "delighted" Harris signed onto his bill. He also set up a series of rhetorical questions and answers describing why he supports a plan that would address the over 27 million Americans who are uninsured, as well as the many more he considers under-insured.

"Do I believe that every American should have health care as a right?" asked Sanders. "I absolutely do."

"Do I believe that we have the most wasteful, expensive, and dysfunctional health care system of any major country on Earth?" he continued. "Yes, I do. Do I believe that we're spending almost twice as much per capita as any other country? Yes, I do. Do I believe that the function of insurance companies is to make as much money as they can at the expense of the American people? Yes, I do. Do I believe that the current Medicare system is a good system? I do. Do I believe it should be expanded to cover all Americans? Yes, I do."