As lawmakers in a nearby hearing room debated last month whether to support her legislation to expand Medicaid, Gov. Laura Kelly of Kansas dared the state’s Republican House speaker to hold a vote.

“If he thinks he can kill it, bring it,” Ms. Kelly, a soft-spoken moderate Democrat, said in an interview in her sprawling office suite at the State Capitol in Topeka.

The next morning, in his own office off the House floor, Speaker Dan Hawkins showed no sign of yielding. He described Medicaid expansion as “almost like the greatest Ponzi scheme ever devised.” The same day, a House committee voted against sending Ms. Kelly’s bill to the floor, derailing the proposal — at least for now.

The standoff between Ms. Kelly and Mr. Hawkins represented one fight in a fierce political battle playing out in several state capitals over the future of Medicaid, the health insurance program for the poor. In Kansas and in a handful of Republican-controlled states in the South, supporters of expanding the program under the Affordable Care Act have renewed their efforts to overcome longstanding Republican opposition, generating a sense of headway.

Yet neither Ms. Kelly nor backers of Medicaid expansion elsewhere have managed to advance legislation far enough to become law, a reflection of the continuing political power of conservative ideas about the nature of government-subsidized coverage and the people deserving of it.

“It’s really the fundamental moral question of where the safety net should be,” said Ty Masterson, the Republican president of the Kansas Senate and a longtime opponent of expansion. “And the safety net should be on the frail and elderly and on the disabled and all the low-income mothers and children.”

The state-level clashes over Medicaid, which is jointly financed by the federal government and the states, could have major implications for hundreds of thousands of low-income Americans, and the debate over the program’s future is unfolding with the 2024 presidential campaign as a backdrop.

In his campaign for re-election, President Biden has highlighted his administration’s work to safeguard the Affordable Care Act. Former President Donald J. Trump, the likely Republican nominee, has threatened the health law in recent months, without detailing his own plans on the subject. Health policy experts have said that a second Trump administration could push for the block-granting of Medicaid, or allowing states to cap the amount of money they spend on the program.

Kansas is one of just 10 states that have not expanded Medicaid under the Affordable Care Act, which allowed adults with incomes up to 138 percent of the federal poverty level, or about $43,000 a year for a family of four, to qualify for the program. All of Kansas’s neighbors have adopted the expansion, three of them — Missouri, Nebraska and Oklahoma — through ballot initiatives in recent years.

Those currently eligible for KanCare, as Kansas’s Medicaid program is known, include children, parents, pregnant women and the disabled. The income limit for many adults to qualify is 38 percent of the poverty level, or roughly $12,000 a year for a family of four. As a result, about 150,000 people fall into what is known as the coverage gap, with incomes too high to qualify for Medicaid but too low to be eligible for a heavily subsidized plan through the federal Affordable Care Act marketplace.

Ms. Kelly and other supporters of Medicaid expansion in Kansas have pressed their case for years. In 2017, the Legislature approved a bill to expand the program, but it was vetoed by the Republican governor at the time, Sam Brownback. Ms. Kelly, who was then a state senator, campaigned on Medicaid expansion in her successful bid for governor the next year and in her re-election bid in 2022.

In her latest attempt to persuade lawmakers to get on board with expansion, Ms. Kelly altered her approach. She unveiled an expansion bill in December that includes a work requirement, offering Republicans a concession she thought could bring them around.

“I tried everything else and it didn’t work,” she said. “I wanted to take it off the table as an excuse.”

At the State Capitol last month, House and Senate lawmakers held two hearings about Medicaid expansion — the first on the topic in four years, giving supporters of the legislation a sense of progress. Both hearing rooms were so crowded that visitors were forced to listen from the hallways or file into overflow rooms.

There have also been signs of movement in Republican-controlled states in the South. In recent months, Republican leaders in Alabama, Georgia and Mississippi have expressed new openness to expanding Medicaid. House and Senate lawmakers in Mississippi approved different Medicaid expansion plans in recent weeks despite intense lobbying from the state’s Republican governor, Tate Reeves, who has vowed to veto any bill that reaches his desk.

“There’s momentum,” Ms. Kelly said.

There have also been setbacks. The same day that lawmakers in Topeka stopped Ms. Kelly’s bill from advancing to the floor, a similar measure in Georgia died in a Senate committee. Mr. Masterson, the Kansas Senate president, argued that the resistance in his state and elsewhere showed that momentum was heading the opposite way.

Mr. Masterson and other opponents of Medicaid expansion have argued that its long-term costs to state budgets are too severe. Supporters have said that the economic rationale is obvious, as the federal government covers 90 percent of the cost. A 2021 pandemic relief package sweetened the deal even further for states that have yet to expand.

Ms. Kelly said that expansion would benefit Kansas’s economy and create thousands of health-related jobs. Officials from hospitals and community clinics in the state see expansion as a potential lifeline for rural providers under financial strain.

Benjamin Anderson, the chief executive of Hutchinson Regional Healthcare System, a rural community hospital outside Wichita, told lawmakers at last month’s hearings that as a lifelong Republican, he opposed the Affordable Care Act. But the state health system’s challenges in picking up the costs of treating the uninsured had persuaded him to support expanding Medicaid. He noted that his hospital had to eliminate 80 jobs last year.

“The next generation of physicians wants to work in a situation where they don’t have to think about how people pay for care,” he said.

The House speaker, Mr. Hawkins, who previously owned a health insurance agency, dismissed the idea that Ms. Kelly’s bill, with its work requirement, could sway voters. No matter the legislation, he said, expansion would bloat the state budget and ask taxpayers to pay for the medical needs of healthy adults who could be working and on employer or marketplace plans.

“We all are supposed to provide them something that they don’t even care enough about to go work and get?” Mr. Hawkins asked. “What happened to our idea in this society that we should be self-supporting, especially if we’re able-bodied?”

The Kansas Health Institute, a nonpartisan research group, has estimated that about 70 percent of those who would become eligible for Medicaid under expansion are working.

One of those who would potentially qualify is Stephen Zook, an uninsured restaurant server in rural Buhler, Kan., who makes around $15,000 each year and falls into the coverage gap in Kansas. He said that he could not see a therapist for depression and other mental health needs, and that he had not been able to pay a roughly $2,000 medical bill he received for a heart monitor last year.

“It’s definitely not people who are lazy,” he said. “It’s people trying to make their lives better for themselves. I’m trying to pull my bootstraps up as many times as I can. And it’s still not enough to get the coverage that I need.”

Melissa Dodge, a single mother of four in Derby, Kan., who works part time as a restaurant hostess and is also stuck in the coverage gap, said she was struggling to get by as she tended to the complex medical needs of her daughter and everyday tasks like school drop-offs.

Her doctor is careful not to order lab tests for her because of the potential for unaffordable costs, Ms. Dodge said.

“It’s a massive source of anxiety,” she said of not having health insurance. “There’s a fear to it that I refuse to allow to run my life. But it’s there. And I can’t not acknowledge it.”

Mr. Hawkins conceded that the politics of the Affordable Care Act had shifted with fading Republican opposition to the law, leaving it as a less potent issue to campaign against. “I just don’t think it has a punch that it once had,” he said.

Ms. Kelly said that if supporters of expansion failed in the current legislative session, they would test the issue on the campaign trail this year. “This will be the No. 1 election issue,” she said.

Ms. Kelly predicted that opponents of expansion were fighting a losing battle.

“They’ve painted themselves into a corner,” she said. “And I think they’re having a hard time finding a face-saving way of getting out.”