But Democrats on the committee pointedly noted that the government already runs a big portion of health care in the United States, in the form of Medicare and Medicaid.
"The main knock is that its government run. Medicare is government run and people like it very much," Schumer said, telling Grassley that if he opposes a public option, what he's saying is "we should have no Medicare because it is a government run plan."
Grassley shot back that Medicare is not at issue because it is now part of the "fabric of our society" and he said that despite the existence of private plans that have bundled Medicare benefits for seniors, the private market cannot exist alongside a public option because "the government isn't a fair competitor. It's a predator."
In another part of Washington, filmmaker Michael Moore said that Obama should "hit the reset button" and call for a single-payer, Medicare for all, health-care system.
At a minimum, Moore -- whose new movie, "Capitalism, A Love Story," premiers in Washington tonight -- said Democrats better support the creation of a public health insurance option that is open to all Americans.
Speaking at a news conference, Moore said if Democrats don't back a public health insurance option open to all Americans, liberals will sit on their hands in 2010 and Democrats could lose control of Congress.
Meanwhile, members of the House are also facing their own issues on a health care bill. House Speaker Nancy Pelosi, D-Calif., and the Democratic leadership are struggling to meld their three health care bills in a way that will get the 218 votes they need to pass a bill.
The thorny point for them are the so-called "majority-makers" -- Democrats elected in 2006 and 2008 from districts that voted for George W. Bush in 2004 and/or Sen. John McCain, R-Ariz., in 2008. There are 84 such Democrats and in this political environment, a vote for a health care overhaul bill may be the political kiss of death for many.
For most of these Democrats, the key issue is not the public option, but costs. The House Democrats' bill, as proposed this summer, would cost more than $1 trillion in the next 10 years, according to the nonpartisan Congressional Budget Office.
The other issues House leaders are struggling with are abortion and immigration. Without alienating progressives, they want to keep the promise that no federal money in an overhaul will go to fund abortion or illegal immigrants.
House majority leader Rep. Steny Hoyer, D-Md., was vague today on the timeline for a House bill. Hoyer told reporters that lawmakers still need to work out what's going to be in and out of the bill and how to pay for it.
He said that the bill will emerge during the month of October but he didn't say when lawmakers would vote on it. Hoyer added that there's no deadline to bring legislation to the House floor, and that it will come to the floor when it is ready.
ABC News' Huma Khan, Teddy Davis, Dean Norland and Elizabeth Gorman contributed to this report.