"Let's start taking test votes on the different elements" and "start making the political points" for "what is going to be a more rational health care system that actually works," Johnson said in an interview. "There is enough resistance and probably recognition" among Republicans that the Senate is likely to move slowly on repeal, he said. "It sounds like President-elect Trump is kind of weighing into it as well, saying, be a little careful here -- we repeal it, it's ours."
Other Republicans are still supporting their leaders' strategy. Sen. Tim Scott of South Carolina said that "if there is a vote to repeal Obamacare I'm going to vote yes," but he wants a three-year delay in implementation for a "comfortable landing." Others also support a swift repeal vote.
"You have to replace it once you repeal. There's a good case to be made to have a trigger that triggers the replacement so everyone knows it's coming. But having them both together is not a necessity," said Sen. Johnny Isakson of Georgia.
Outside groups are also putting pressure on the House GOP to act quickly.
"Political risk, real or imagined, should not be used as an excuse for members of Congress to avoid doing the job they signed up for in November," James Wallner, a vice president at the conservative Heritage Foundation, wrote in an op-ed. "No one said it was going to be easy."
Sen. Lamar Alexander of Tennessee, the chair of the health committee, indicated that the process of voting to replace Obamacare would take until the summer to complete, and said Republicans are "going to do this step by step."
"We need to carefully reform and replace Obamacare, and when it takes effect, we can finally repeal Obamacare. And so we've got to get the right sequencing on this," Alexander told reporters. "In my view we need to cast most of our votes on that before summer time. It'll probably take two or three steps and then it'll probably take two or three years to implement it over time."
Another source of GOP division is whether a replacement should insure as many people as Obamacare does, in order to avoid the political fallout of throwing people off coverage. Some Republicans, such as Sen. Shelley Moore Capito of West Virginia, say they want their party's alternative to cover at least as many Americans. Others demur on the question: "We'll see. We don't know yet," Alexander said.
The backers of the new amendment insist that they remain committed to the goal of dismantling President Barack Obama's health care law.
"This amendment will ensure that we move forward with a smart, responsible plan to replace the law as quickly as possible," Portman said.
Murkowski said it's simply "common sense" that repeal and replace happen simultaneously.
"I remain committed to repealing the Affordable Care Act, and I am equally committed to ensuring that all Alaskans and Americans, especially the most vulnerable among us and those in rural communities, have access to affordable, quality health care," Murkowski said.
( Laura Litvan, James Rowley, Erik Wasson, Billy House, Anna Edgerton and Toluse Olorunnipa contributed to this report.)
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