On Monday, more senators said they agreed with a delay on the tax front, including Sen. Mike Rounds of South Dakota.
John Cornyn, the No. 2 Senate Republican, told reporters that the taxes used to subsidize insurance for millions of Americans could be dealt with later this year in a larger tax overhaul.
Republicans senators are also grappling with the risks of repealing the law before a replacement is ready.
Sen. Ron Johnson of Wisconsin said he wants to see a "pretty darn specific" proposal to replace Obamacare before it's repealed.
"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.
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," said Alexander.