House Democratic leaders still do not have enough votes to pass health care reform, the chamber's top vote counter said Sunday, even though the administration is aiming to have the bill passed this week. 

The reality check came from Rep. James Clyburn, the House Democratic whip. 

"No, we don't have them as of this morning, but we've been working this thing all weekend," Clyburn, D-S.C., said. 

Despite the challenge of corralling wavering Democrats, Clyburn joined with other Democratic officials in saying he was confident the measure would pass, echoing comments from House Speaker Nancy Pelosi Saturday. 

Democratic leaders want the House to pass the Senate-approved version of the bill before moving on to a package of changes. That package, which is already under consideration, would be taken up in the Senate as a "reconciliation" bill, meaning the Senate would be able to approve it with just 51 votes -- Democrats turned to the legislative tactic after Republican Scott Brown won the Massachusetts Senate election in January, breaking the party's 60-vote filibuster-proof majority. 

More On This...

The administration is pressing hard for every piece of this delicate sequence of events to fall into place. 

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

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

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. 

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 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, 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, who stopped paying for her insurance in January, was being admitted to the Cleveland Clinich after being diagnosed last week with leukemia.

Gibbs said Canfield 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. 

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." 

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. 

The Associated Press contributed to this report.