Surveys indicate that calls to abandon the bills, the product of six months of tough compromise among Obama's ruling Democrats, could boost support for Republicans as they look toward November's congressional elections.
Democrats reject that approach, although they have not yet made clear how they will go forward on healthcare reform after losing a critical supermajority in the Senate in January that allowed them to override Republican blocking manmaneuverseuvres.
With about 46 million people lacking health insurance, the surveys show a high level of frustration as Americans watch the abortive reform process at a time of economic crisis and high unemployment that has eroded support for Obama.
Kathy Boatman, a committed Republican who is unemployed and lives in Arizona's Phoenix Valley, wants the bills thrown out. I'd like to see them (Democrats) bin it completely, she said.
She was speaking at a meeting of activists of the Tea Party movement, which advocates small government, low taxes and fiscal restraint and whose members portray Obama as seeking a government takeover of the healthcare system.
I think he should either keep pushing his (comprehensive) package or wait another year. I don't think a compromise loaf is better than no loaf at all, said San Francisco attorney Britt Cibulka, 40, who voted for Obama in 2008.
Obama is under pressure to produce results on this key domestic priority ahead of the elections.
He invited Republicans to bring their best ideas to a televised bipartisan meeting on February 25, but his opponents say the meeting will have limited value if its starting point remains the stalled bills.
It really is time to scrap the bill and start over, John Boehner, Republican leader in the House of Representatives, said on Tuesday after a meeting at the White House.
Polls show many people hungry for bipartisan compromise.
A Washington Post/ABC News poll released this week found two-thirds of Americans want to see Democrats and Republicans working to pass comprehensive healthcare reform.
It showed big support for key elements of the legislation, with 80 percent backing a ban on insurance company limits on people with pre-existing health conditions and 72 percent favoring employers being obliged to pay for health insurance.
But 60 percent said the legislation in the Senate and House bills, which have significant differences, was too complicated. Almost the same percentage balked at the costs.
The key to understanding public sentiment could lie in the way the issue is framed, said Scott Keeter of the non-partisan Pew Research Center.
We know from extensive polling last year that majorities of the public favor many of the key elements in the bills that both houses of Congress passed, said Keeter.
Whether or not people want to 'start over' or not depends on what is meant by 'start over.' If what you offer in a poll is (1) current bill, (2) entirely new bill, (3) no reform, you'll get a majority out of the last two options, given the level of support for the current bill, he said.
For many voters the debate directly affects their economic and financial health.
Chitra Staley in Cambridge, Massachusetts, runs a small financial planning firm for wealthy people and pays the full premiums for healthcare insurance for her two employees.
A lot of people are talking about what should he (Obama) do. My opinion frankly is that although I'd love to see the healthcare bill pass so that everyone has coverage, it's practically impossible now, she said.
Staley said given the circumstances she would like Obama to push forward with incremental changes. Get bits and pieces passed every year, she said. Massachusetts already has a 98 percent insured rate under its 2006 healthcare reform bill.