But President-elect Donald Trump says any delay is unacceptable.
According to the Times, Trump said "he would not accept a delay of more than a few weeks before a replacement plan was voted on."
Congressional Republicans, however, are nowhere close to proposing a plan to replace the sweeping healthcare law, let alone having something in place that can be voted on, withstand a Senate filibuster and be sent to the White House for Trump's signature.
These comments from Trump could put Republican leaders in a bind, as they have already been fielding widespread concerns from rank-and-file members about moving too fast to dismantle Obamacare when there is little consensus on an alternative.
Trump's interview also appears to suggest the incoming President's lack of understanding about the process that is already unfolding on Capitol Hill to repeal large parts of the law -- a first procedural vote is expected some time this week, but a final repeal bill was not expected to be voted on for weeks or months.
The President-elect's direction to Republicans was published in the Times just hours after Ryan said that GOP leadership now hopes to tackle "repeal and replace" simultaneously -- a dramatic turn of events from the original goal of quickly repealing the healthcare law and then spending years standing up a new system.
"It is our goal to bring it all together concurrently," Ryan told reporters after a closed-door meeting with House GOP members.
Ryan emphasized that it would be a complicated and multi-pronged effort. "We'll use every tool at our disposal through legislation, through regulation, to bring replace concurrent along with repeal so we can save people from this mess," Ryan said.
Republicans on Capitol Hill are united on getting rid of President Barack Obama's signature healthcare law. The House and Senate are expected to vote this week on a budget resolution -- dubbed by Republicans as the "repeal resolution" -- that will be the first of two steps in rolling back major parts of Obamacare.
"He's saying we can't wait two years, we would all agree with that," said Rep. Chris Collins, R-New York, and a member of Trump's transition team, said of Trump's comments. "It's also, given this legislative body, not going to be in two weeks."
Collins said Trump is a CEO and using that mindset to make a decision and wants Republicans in Congress to do the same.
"That's where Donald Trump the CEO versus a legislative body that tends to procrastinate when they don't want make a decision will be butting heads some," he said.
"He wants us moving it through so we can put it on his desk right away," Collins added. "I'm not reading it literally literally."
Alarm bells from GOP lawmakers
Republicans came back to Washington with a plan to quickly repeal the law while working on a replacement, but as they confront the practical impact of wiping away much of the current health care system, Republicans are also anxious to quickly reassure the public that they have a plan for what would come in its place.
With the budget resolution vote just days away, alarm bells are ringing out across Capitol Hill.
Senate Republicans have grown increasingly vocal, airing concerns that the party may be moving too fast to repeal a law that covers some 20 million people; in the House, GOP members ranging from conservatives to moderates are also joining in.
These worries spilled out into the open during a Tuesday morning House GOP conference meeting, where members shared their unease about the upcoming budget resolution vote.
Rep. Mark Meadows, the chairman of the conservative House Freedom Caucus, said the group will not take an official position on the vote until they get more reassurances from leadership.
"We just would like some more specifics in terms of repeal and replacement aspects of the Affordable Care Act, where we're going with it," Meadows said as he left the meeting. "Hopefully, what we will see and we're hopeful that leadership will give us some more specifics in the next 24 to 48 hours."
"I ran for office on repeal and replace. I certainly think we should advance replacement aspects as quickly as possible," New Jersey Republican Rep. Leonard Lance told CNN.
Lance said he was willing to support the procedural steps to begin unwinding the law but stressed that he wants a clear timeline for those covered under Obamcare's Medicaid expansion so patients know where to get coverage under a new program.
Kentucky Rep. Andy Barr suggested that there were tensions in Tuesday's meeting between members who were emphatic that a repeal and replace vote happen simultaneously, and others who stressed that it's simply impossible to move toward repeal until the Republicans first pass the budget resolution.
"There's some folks who have expressed a desire to do everything at once but I don't think they fully understand that if we don't take these first steps, we're never going to get to the next step," said Barr, a member of the Republican Study Committee, which introduced its own replacement plan last week.
Barr said he will vote in support of the budget resolution and that he expects a "very large majority" of his conference to do the same.
Ryan and other leaders discussed ideas for adding replacement into an upcoming budget reconciliation bill -- the vehicle through which Republicans plan to roll back Obamacare. But the speaker admitted that they haven't decided yet how much of the law they can dismantle or replace with this approach, pointing out that there are rules limiting what the budget reconciliation bill can include.
"We will pass as much as we can through whatever vehicle we've got and then we will pass all the other things through regular order outside of reconciliation that show you the full scope of what a real replacement effort will look like," Ryan said.
Florida GOP Rep. Dennis Ross told reporters that the hope is to "do as much replacement as we can get away with in the reconciliation package" before throwing the ball back to the Senate.
He said Republicans will focus on measures related to tax and spending, including provisions to promote health savings accounts, remove the financial penalty for those who don't enroll in Obamacare.
Republican leaders also continue to insist there will be a smooth transition to a new system and that they won't abruptly cancel insurance policies.
House GOP Conference Chair Cathy McMorris Rodgers took those reassurances one step further on Tuesday, saying: "Let me be clear: no one who has coverage because of Obamacare today will lose that coverage."
Others stressed the need to begin the process now to repeal the law, but seemed comfortable with allotting plenty of time for the rest of the process to play out.
"There is a recognition that Obamacare took a long time to construct," Illinois Republican Rep. Peter Roskam said, "and it's similarly going to take a long time to deconstruct."
Trump as businessman, not legislator
Republicans working on the healthcare issue downplayed Trump's comments, which came after the closed-door conference meeting, pointing to the difference between the President-elect's perspective as a businessman rather than a political leader.
"You can't repeal it and replace it quickly - that's an oxymoron," Collinstold reporters. "We can repeal it quickly and replace it in a timely fashion."
Many House GOP members are reluctant to vote for the budget resolution on Friday until some of their questions about the plan ahead are cleared up.
"We should not vote until we see the plans and well in front, plenty of time on what the fix is," House Rules Committee Chairman Pete Sessions said Tuesday.
He said members are "entitled" to have details on the replacement "before we do anything." Sessions said GOP leaders still trying to figure out the next steps in terms of the budget vote slated for Friday.
North Carolina Rep. Richard Hudson hadn't read the Times story, but he essentially agreed with Trump's message to proceed quickly. "I think that's where the American people are and we are working through the process," he said. "Does that means weeks or days, does it mean hours -- we are working through the timing."
Asked if Trump doesn't understand the process, Hudson said, "the guy's not a politician, he's a businessman."