Lawmakers' hands may be tied because of the method they are using to pass the legislation: A budget resolution allows the repeal bill to pass with 50 votes and bypass a Democratic filibuster, but it can cover only budget issues.
"We're going to do as much as we can," Cornyn said. "The fact of the matter is, it's going to take a while for us to make the transition from Obamacare to the alternative."
He acknowledged worries from senators in his party "about the meltdown on Obamacare happening while we're trying to make the longer term replacement," but said the two chambers would find an agreement.
"I'm convinced, like so many things around here that seem a little confused at the beginning, that if put our minds to it we'll come up with a reasonable solution," Cornyn said.
Other senators downplayed the conflict, saying lawmakers had time to hammer out the details.
"I worry more about what we told the American people in the last election: 'Repeal and replace,'" said Sen. John McCain, R-Ariz.
Democrats, who ahead of the Thursday vote hope to force Republicans to vote on a series of health care-related amendments that could later be used against them in campaign ads, have pledged a public relations assault, urging supporters to call their member of Congress or McConnell's office to protest the move to repeal.
"We have to have a national movement here," Senate Minority Leader Chuck Schumer, D-N.Y. told members of progressive groups on a call late Monday. "This should not be just something a bunch of senators are talking about on the floor. We ain't resting until the move to repeal ACA is dead -- it has a dagger through its heart and will never be revitalized."
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