WASHINGTON — The House’s chief Democratic headcounter said Sunday he hadn’t rounded up enough votes to pass President 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 its 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 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 chairmen, 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 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 today to Strongsville, Ohio, home of cancer patient Natoma Canfield, who wrote the president 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.

Canfield’s sister was scheduled to introduce Obama at the 1 p.m. event.