WASHINGTON – The House's chief Democratic headcounter said Sunday he hadn't rounded up enough votes to pass President Barack Obama's health care overhaul heading into a make-or-break week, even as the White House's top political adviser said he was "absolutely confident" in its prospects.

The administration gave signs of retreating on demands that senators jettison special home-state deals sought by individual lawmakers that have angered the public.

White House spokesman Robert Gibbs predicted House passage this week, before Obama travels to Asia, a trip he postponed to push for the bill.

"This is the week where we will have this important vote," Gibbs said. "I do think this is the climactic week for health care reform."

Political strategist David Axelrod said Democrats will persuade enough lawmakers to vote "yes." The House GOP leader, Ohio Rep. John Boehner, took up the challenge, acknowledging Republicans alone can't stop the measure, but pledging to do "everything we can to make it difficult for them, if not impossible, to pass the bill." Republicans believe they may get help from Democrats facing tough re-election campaigns.

Axelrod said it will be a struggle, taking aim at insurance industry lobbyists who "have landed on Capitol Hill like locusts" and Republicans who see being on the losing side of the vote as a political victory.

"I am absolutely confident that we are going to be successful. I believe that there is a sense of urgency on the part of members of Congress," given recent news about insurance plan rate increases, Axelrod said.

A dose of reality came from Rep. James Clyburn, the third-ranking House Democrat and main vote counter. "No, we don't have them as of this morning, but we've been working this thing all weekend," said Clyburn, D-S.C.

Clyburn said he was confident the measure would pass, echoing comments from Speaker Nancy Pelosi, D-Calif., on Saturday.

Axelrod also indicated the White House was backing down on an attempt to get senators to rid the legislation of a number of lawmakers' special deals.

Taking a new position, he said the White House only objects to state-specific arrangements, such as an increase in Medicaid funding for Nebraska, ridiculed as the "Cornhusker Kickback." That's being cut, but provisions that could affect more than one state are OK, Axelrod said.

That means that deals sought by senators from Montana and Connecticut would be fine — even though Gibbs last week singled them out as items Obama wanted removed. There was resistance, however, from two powerful committee chairman, Democratic Sens. Max Baucus of Montana and Chris Dodd of Connecticut, and the White House has apparently backed down.

"The principle that we want to apply is that are these — are these applicable to all states? Even if they do not qualify now, would they qualify under certain sets of circumstances," Axelrod said.

That's the argument made by aides to Baucus and Dodd. The measure to give Medicare coverage to asbestos-sickened residents of Libby, Mont., could apply to other places where public health emergencies are declared — even though Libby is the only place where that's happened so far. Dodd's deal would leave it up to the health secretary to decide where to spend $100 million for construction of a hospital, though Dodd has made clear he hopes the University of Connecticut would be the beneficiary.

Trying to increase public pressure on Congress to pass the legislation, Obama planned to travel on Monday to Strongsville, Ohio, the home of cancer patient Natoma Canfield, who wrote the president that she gave up her health insurance premium after it rose to $8,500 a year. Canfield is a self-employed cleaning worker who lives in the Cleveland suburb.

Gibbs said she had to decide between keeping her health insurance or her house, and chose to keep her house.

Boehner said Democrats never made a serious attempt to incorporate GOP ideas in the measure, saying they took only "a couple of Republican bread crumbs and put them on top of their 2,700-page bill."

The legislation would provide health insurance to tens of millions who currently have none and would ban insurance companies from denying coverage on the basis of pre-existing conditions. It would require most people to obtain insurance and would subsidize premiums for poor and middle-income Americans.

The health care bill appeared close to passage in January, before Republican Scott Brown won the Massachusetts special election to fill the seat of the late Edward M. Kennedy, which cost the Democrats a fillibuster-proof majority in the Senate.

Since then, the White House and Democrats have tried to rescue the effort. Besides Republican opposition, Democrats still are face resistance in their own party from anti-abortion lawmakers worried about how and whether insurance plans should pay for abortions. The bill needs 216 votes to clear the House.

Axelrod was on ABC's "This Week," NBC's "Meet the Press," and CNN's "State of the Union." Gibbs appeared on "Fox News Sunday" and CBS' "Face the Nation." Clyburn was on NBC and Boehner on CNN.

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