Washington (CNN) Donald Trump's presidency is off to an explosive start: A clash with the media over crowd size on Inauguration Day; the theatrical unveiling of a Supreme Court nominee; and the launch of an immigration crackdown that sparked nationwide protests.

But across town on Capitol Hill, Republican lawmakers have been working away at the monumental task of overhauling Obamacare in the opposite fashion -- quietly, slowly and with little fanfare. And they're starting to be more open about acknowledging that the Affordable Care Act may never entirely disappear.

GOP Rep. Greg Walden, the chairman of the House Energy and Commerce Committee, said in an interview Wednesday that it is time to do away with the notion that there is a GOP Obamacare "mega-bill" in the works.

"There isn't," said Walden, whose committee has significant jurisdiction over healthcare. "We're looking at fixing this mess a brick at a time. Piece by piece. Taking our time to get it right."

That sentiment is far from the promises Trump and many other Republicans have made in the past to swiftly and decisively repeal and replace Obamacare. Other key Republicans echoed Walden's caution.

Sen. Orrin Hatch, the chairman of the Senate Finance Committee and another key player in the health care debate, said Wednesday that as much as he supports repealing Obamacare, Republicans should also "try and repair the law."

"I'm for repealing it and starting over, but you can certainly look at the good things that may be part of the law," Hatch said. "There are some good things that we would put in any bill."

Senate Health Committee Chairman Lamar Alexander described the GOP's task at hand as first "repairing the damage that Obamacare has done," and then "repealing the parts of Obamacare that caused the problem."

"We're going to have outright repeal, but it may take place at different times but on different schedules," Alexander told CNN.

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There's no doubt that overhauling President Barack Obama's legacy accomplishment remains an urgent political priority for the GOP. Trump explicitly ran on undoing the law as a presidential candidate -- the same way that Republican officeholders across the country rallied their supporters with that promise in every election cycle since 2010.

But despite the impassioned rhetoric from years past -- and despite being armed with the powerful advantage of controlling both chambers of Congress and the White House -- Republican are increasingly eager to cast the process of dismantling Obamacare as a long-haul marathon.

This tempered rhetoric could anger some segments of the GOP base, which for years has heard Republican candidates chant "repeal and replace Obamacare" as a snappy political slogan.

Sen. Brown: Republicans are getting nervous

"I'm hearing more and more quiet conversations. They're expressing public bravado about 'repeal and replace' and they're nervous and jittery and they know what they're doing is going to hurt millions of people," Democratic Sen. Sherrod Brown said of his GOP colleagues in an interview with CNN.

The perception of working with Democrats to make incremental changes to Obamacare, Brown added, would amount to political suicide for Republicans.

"That would make their base mad," he said. "They've been singing this song for six years and their base (will say), how come you're not doing this?"

Multiple tracks

In the clearest sign that it is simply impossible to quickly and entirely repeal and replace a health care law that covers 20 million people, the party is currently moving on multiple tracks and taking a piecemeal approach to rolling back the law.

Its most immediate task is to pass a bill to roll back significant portions of the Affordable Care Act through the budget reconciliation process. That legislation is being written by policy aides spanning several key congressional committees with guidance from GOP leadership, with an eye towards a late-February to March timeframe.

The writing of that repeal bill became more complicated when a broad spectrum of rank-and-file Republicans began to express concern in recent weeks that the GOP is moving too fast on repealing Obamacare. In part to ease those concerns, GOP leaders agreed to explore ways of including some replacement measures in that legislation.

And for the first time this Congress, lawmakers have started to debate legislation to make changes to Obamacare.

On Thursday, a House subcommittee will examine a group of bills aimed at ensuring those with pre-existing conditions are not denied coverage -- one of the most popular aspects of Obamacare -- as well as lowering premiums.

Two bills address insurers' requests to tighten the rules for signing up outside the standard open enrollment period and reduce the grace period consumers have to pay their premiums while still maintaining coverage. Another would allow insurers to charge older enrollees more compared to their younger consumers.

In a hearing Wednesday, members of a House Energy subcommittee considered several bills to tighten Medicaid rules, including ending benefits for lottery winners and closing a loophole that allows married couples to shelter assets in order to qualify for the program.

HHS nominee Tom Price still waiting

GOP lawmakers are also counting on one colleague in particular to help with some of the heavy-lifting: Rep. Tom Price.

Price, Trump's nominee to lead the Department of Health and Human Services, is expected to be confirmed by the Senate in a matter of days following a dramatic boycott from Democrats who are unhappy with Price's vetting.

As head of the HHS, Price will have a broad mandate to carry out an Obamacare executive order that Trump signed on his first day in office. The Georgia congressman will also have a slew of other administrative tools at his disposal to dilute and weaken the law.

"Some of that will happen over there (at HHS). Some of that we will have to legislate. Some of that will be in reconciliation," Walden said, describing the many ways in which Republicans are tackling Obamacare.

The congressman, who met a group of centrist Republicans and Democrats on Wednesday to discuss Obamacare, acknowledged that a part of the challenge will be appeasing lawmakers across the political spectrum.

"You know there's a wide range of views on what we should or shouldn't do," Walden said.