The Senate rejected a proposal Wednesday that would have repealed major parts of the Affordable Care Act, but Republican leaders were growing more confident about their chances of passing a more modest overhaul of the health-care law later this week.

Republicans appeared to be ­coalescing around a “skinny repeal” that would abolish the individual and employer insurance mandates and perhaps just one tax in an attempt to sustain their seven-year quest to unwind President Barack Obama’s health-care law. But even if they succeed — and start negotiations with the House — they will face significant obstacles in accomplishing anything more substantial.

Top Republicans such as Sen. John Thune (S.D.), the chamber’s third-ranking Republican, said that although leaders have not yet found “the sweet spot” between conservatives and centrists, they had picked up support for a more modest plan because it did not include deep cuts to Medicaid. Some Republican senators were simply open to any legislation that could keep alive the roller-coaster push for an overhaul.

“We’re edging closer and closer” to getting 50 votes for a bare-bones plan, Thune said. He said leaders were betting that some Republicans who defected on votes this week would feel more pressure to support any bill that emerged from negotiations with the House to face a final vote in the Senate.

“Voting on something at the end of the process when it’s the only train leaving the station . . . I think that’s a different vote for a lot of people,” he said.

View Graphic Which health-care plans the Senate rejected (and who voted ‘no’)

More than half a dozen centrists from states that expanded Medicaid under the Affordable Care Act objected to the original Senate draft GOP leaders crafted last month.It would have cut the program for low-income Americans by $772 billion over 10 years and curtailed its long-term growth rate.

Yet even if Republicans agree on a minimalist plan to alter the ACA, uniting with their House colleagues to enact a bill would be far more challenging. On Wednesday — even before the skinny repeal came up for a vote — some House conservatives were calling it untenable.

Rep. Mark Meadows (R-N.C.), chairman of the House Freedom Caucus and a key player in negotiating the House-passed bill, told reporters recently that a skinny repeal would be “dead on arrival” in the House and that a conference committee would have to be convened to work out a compromise.

Senate Majority Leader Mitch McConnell (R-Ky.) overcame serious opposition from his rank-and-file members to begin debate on health care — a prospect that seemed dim just last week. President Trump has taken to Twitter and made public statements challenging Senate Republicans to support an overhaul or take ownership of the ACA’s failure.

But in two votes within 24 hours of each other, lawmakers rejected differing approaches to rewriting the landmark 2010 law. The open voting process — which is likely to drag on for the rest of the week — has laid bare the fact that Senate Republicans haven’t been able to find a comprehensive replacement for the law they have relentlessly lambasted.

Republicans on Wednesday lacked answers for how or even whether they can break their gridlock by simply extending their endeavor, but they appeared determined to press ahead.

“I think it’s a good idea to start with what we agree on and see how big we can get the bill from there,” said Sen. Rand Paul (Ky.), who has pushed for a repeal of the law and has repeatedly clashed with GOP leaders.

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Sen. Lindsey O. Graham (S.C.) said that a scaled-back bill “is not a solution to the problem” the American health-care system is facing, but that there did not appear to be another option.

GOP leaders have little room to navigate when it comes to crafting a bill, as just three defections within their ranks would deprive them of the 50 votes they need to pass legislation with the assistance of Vice President Pence, who can break a tie.

And in each of the two most important votes the Senate has cast since taking up the bill, at least 13 percent of Republicans defected to join Democrats in opposition.

“This certainly won’t be easy. Hardly anything in this process has been,” McConnell said on the Senate floor Wednesday.

In an effort to muster enough votes for a narrow bill, GOP leaders suggested that even some proposals that have died in the Senate could come up again once they enter negotiations with the House.

Senate Majority Whip John Cornyn (Tex.) said proposals offered by Sens. Rob Portman (R-Ohio) and Ted Cruz (R-Tex.) that were rejected Tuesday as part of a broader rewrite measure could resurface. Graham, meanwhile, said he is willing to go along with the skinny repeal — but only if he is assured that a plan he has offered would be reconsidered.

Portman’s measure calls for adding $100 billion in federal funding to help consumers with out-of-pocket medical costs and allowing states to provide cost-sharing assistance to low-income people who transition from Medicaid to private insurance with a federal tax credit. Cruz’s amendment would let insurers offer health plans on the ACA market that do not provide the full benefits required under the law, as long as they offer at least one option that does.

A total of 57 senators, including nine Republicans, voted against the measure that included both of those provisions. But Cornyn said that passing a skinny repeal would buy time for the Congressional Budget Office to score those two plans, which may be revisited in a conference committee.

Senate Democrats announced late Wednesday afternoon that a preliminary CBO estimate found that 16 million people would lose coverage if Republicans enacted a handful of the policies floated for the pared-down repeal bill. The analysis was based on the assumption that the GOP wants to repeal the individual and employer mandates, end a 2.3 percent tax on medical device manufacturers, ban funding for Planned Parenthood and repeal prevention health funds.

In a sign of how the prospect of a spike in the uninsured rate continues to worry governors, a bipartisan group of 10 of them — including Republicans Brian Sandoval of Nevada and John Kasich of Ohio — urged Senate leaders late Wednesday to work together with governors in developing a new plan and to reject a skinny repeal, which they said “is expected to accelerate health plans leaving the individual market, increase premiums, and result in fewer Americans having access to insurance.”

Senate Republicans hope that once their members are faced with enacting an imperfect measure, or not accomplishing one of their chief legislative goals, they will decide that some progress is better than none.

That sort of thinking prompted Senate Minority Leader Charles E. Schumer (D-N.Y.) to say Wednesday that a skinny repeal is a ­“Trojan horse” that would lead House conservatives to push the plan back to a much more aggressive attack against the ACA.

“There is no such thing as ­‘skinny’ repeal; it’s a ruse to get to full repeal, with all the concomitant cuts to Medicaid and tax breaks,” Schumer said on the floor.

The Senate also voted down a pair of attempts by Democrats to end debate by forcing two Senate committees to review and debate the legislation, and an amendment from Sen. Dean Heller (R-Nev.) affirming support for Medicaid and asking for the Senate to review the program further. Democrats grew frustrated by the spectacle Wednesday evening and threatened to stop offering amendments until GOP leaders released details of the narrow repeal measure they plan to offer.

Meanwhile, the ongoing uncertainty on Capitol Hill sent jitters through the insurance industry.

Joseph R. Swedish, the chief executive of Anthem — the nation’s second-largest health insurer — said on a conference call to review second-quarter earnings that the company is reassessing its participation in ACA marketplaces for next year. Anthem has decided to largely withdraw from the markets in three of the 14 states it participates in, and he said it may stop participating elsewhere unless the markets seem stable.

He cited, in particular, the question of whether Congress and the Trump administration will continue “cost-sharing subsidies” that the ACA provides insurers to help lower-income customers — about 7 million this year — afford deductibles and other out-of-pocket expenses.

Noting that Anthem’s “2018 market footprint” for selling ACA health plans is not fully decided, Swedish said, “If we aren’t able to gain certainty on some of these items quickly, we do expect that we will need to revise our rate filings to further narrow our level of participation.”

The Blue Cross Blue Shield Association warned that even the skinny repeal Republicans now envision could undermine the individual insurance market because it would eliminate the requirement that Americans buy insurance or pay a tax penalty. The measure, which remains subject to negotiation, also would probably eliminate the ACA’s requirement that employers with 50 workers or more provide health coverage, and a medical device tax that generates $19.6 billion in federal revenue over a decade.

“If there is no longer a requirement for everyone to purchase coverage, it is critical that any legislation include strong incentives for people to obtain health insurance and keep it year-round,” the Blue Cross Blue Shield Association said in a statement.

A CBO estimate in December projected that repealing the ACA’s individual mandate would leave 15 million Americans uninsured most years and prompt premiums to rise by 20 percent.

Several Republicans appeared wary Wednesday of moving too quickly to undo the health-care law without a replacement in hand. That proposal was defeated on a vote of 55 to 45, with seven Republican senators — including John McCain (Ariz.) and Lamar Alexander (Tenn.), who chairs one of the key committees that would normally craft a health-care bill — opposing it.

Alexander said after the vote that although he supported an outright repeal in 2015, his constituents could not tolerate that kind of uncertainty now.

“I don’t think most Tennesseans would like the idea of our saying to them, ‘We’re going to cancel insurance for 22 million Americans and then trust Congress to replace it in two years,’ ” he said. “I think most pilots, when they take off, like to know where they’re going to land.”

The mood among Republicans on Wednesday was far from the excitement that some expected to accompany the first votes to fulfill their long-standing promise to repeal the ACA. Instead, they described feeling frustrated and unhappy with their options.

“The mood is nothing,” Sen. Bob Corker (Tenn.) told reporters after Tuesday’s failed vote on the Senate GOP’s original plan. “It’s perfunctory.”

[Analysis: Republicans no longer have a plan to repeal Obamacare, but they’re still trying]

Amy Goldstein and Mike DeBonis contributed to this report.

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