House Republicans emerged from a meeting with Vice President-elect Mike Pence Wednesday pledging to repeal and replace Obamacare in an orderly way, even signaling that Donald Trump is ready and willing to use his pen to stabilize the transition to a new program.

Mr. Pence and GOP troops said the president-elect is working on as-yet-unspecified ways to use “executive authority” to make sure the millions of people who gained insurance under President Obama’s signature overhaul do not see their coverage unravel while Republicans in charge of Congress forge ahead with fast-track repeal without a replacement in hand.

“We’re working on both a legislative and executive action agenda to ensure that an orderly and smooth transition to a market-based health care reform system is achieved,” Mr. Pence said, without specifying which parts of Obamacare might remain in place.

Mr. Trump on Wednesday posted a series of cryptic messages saying Republicans should be “careful” as they plot out their strategy, arguing the Affordable Care Act ushered in horrific premium increases so Democrats should take all the blame for what happens next.

“It will fall of its own weight — be careful!” he said on Twitter.

Repealing the law is the No. 1 priority for Republicans who’ve railed against Obamacare for years but were unable to unwind it, so long as its namesake was in office.

Mr. Pence said voters who elected him and Mr. Trump sent a loud and clear message that they were tired of the law, though he’s mindful of those who are using it.

“We’re talking about people’s lives, we’re talking about families. But we’re also talking about a policy that has been a failure, virtually since its inception,” Mr. Pence said.

Senate Republicans gaveled in Tuesday and started work on a budget that will instruct committees to develop legislation that guts Obamacare on majority votes in either chamber.

The fast-track process, known as budget reconciliation, allows Senate Republicans who hold just 52 seats to bypass a Democratic filibuster, though passing a replacement down the road will take 60 votes.

Republicans who met with Mr. Pence offered scant details on the timeline for repeal, though budget instructions told a quartet of House and Senate committees to report their ideas by Jan. 27, signaling a swift timeline for getting their plans to the floor in the first months of Mr. Trump’s administration.

Replacing the law will be much harder, since it is unclear whether Republicans can reach a consensus on a plan and peel off enough Senate Democrats to overcome a potential filibuster.

Rep. Chris Collins, New York Republican and early Trump supporter, said the party might be able to release a proposal within six months, though other members would not commit to a timeline after the Pence meeting.

GOP lawmakers said they wanted to proceed carefully through committees and make sure they do not muscle the law through the way Democrats flexed their majorities in 2010, when not a single Republican voted for Obamacare’s passage.

“We’re gonna do it right,” Mr. Collins said.

Republicans said Mr. Trump will likely use executive action on “day one” to roll back various parts of Mr. Obama’s agenda, though he will swiftly turn to additional steps to proactively pave the way for a stable Obamacare transition, since insurers will need certainty to price their plans.

“There are market realities to this,” Rep. Mike Coffman, Colorado Republican, said.

Leading Democrats say the GOP will pay a political price in their rush to dismantle President Obama’s signature domestic achievement with a replacement in hand.

While Republicans met behind closed-doors, Democrats huddled with Mr. Obama in another part of the Capitol to plan their defense of the law, which cut the uninsured rate to historic lows by expanding Medicaid in certain states and offering taxpayer subsidies to qualified persons seeking private plans on web-based insurance exchanges.

Mr. Obama declined to take questions after the meeting, opting instead for photographs with young Senate pages, though CNN reported the president told Democrats to be as aggressive in their defense of Obamacare as the hard-right tea party was in condemning it in 2010.

“Look out for the American people,” he advised Democrats as he left the Capitol.

Republicans say Obamacare is collapsing, however, as insurers flee its exchanges and as premiums in the individual market jump by double digits, so it is time to fulfill their campaign promise and provide immediate “relief” from heavy mandates and coverage requirements alongside Mr. Trump, who has publicly backed their plans.

“This law has failed. It’s getting worse,” House Speaker Paul D. Ryan said.

House Republicans laid out the contours of their plan in an election-year blueprint that calls for age-based tax credits and allowing people to shop across state lines to entice people into plans, rather than relying on a government mandate and tax penalties to prod people into coverage. It also suggests state-based, high-risk pools to cover sicker customers who’ve been priced out of the market.

However, key details are missing. For instance, it is unclear how generous its tax credits would be, how many people might be covered and how much the program would cost the federal government, since the party hasn’t coalesced around a plan that can be scored by congressional budget keepers.

Sen. Joe Manchin III, West Virginia Democrat who declined to attend the Obama meeting, said Wednesday’s partisan huddles at the Capitol flouted the will of voters who wanted to see Washington work together.

Yet he chided the GOP for pushing repeal before they settle on a repeal bill.

“I’m willing to look at replacing, repairing, doing anything that we can to make it better, but put something on the table,” Mr. Manchin told MSNBC’s “Morning Joe” program. “I just can’t believe that Republicans will go down this path and just throw it out and say, ‘Trust us. In two or three years, we’ll fix it.’ “

A handful of Senate Republican also have signaled that repealing and replacing the law should come at the same time — a message the American Medical Association echoed in a letter to congressional leaders this week.

“We believe that before any action is taken through reconciliation or other means that would potentially alter coverage, policymakers should lay out for the American people, in reasonable detail, what will replace current policies,” CEO James L. Madara said. “Patients and other stakeholders should be able to clearly compare current policy to new proposals so they can make informed decisions about whether it represents a step forward in the ongoing process of health reform.”

• David Sherfinski and Dave Boyer contributed to this report.

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