Tarren Bragdon, the chief executive of the conservative Foundation for Government Accountability, said that to governors of both parties, federal funds looked like “free money.” By contrast, he said, Republicans in Congress focus on costs to the federal government and believe that the expansion of Medicaid will not be sustainable or affordable in the long term.

When Democrats wrote the Affordable Care Act, they wanted to make Medicaid available to all Americans under 65 with incomes up to 138 percent of the poverty level — or $16,240 for an individual. The federal government pays the full cost for newly eligible beneficiaries from 2014 to 2016 and at least 90 percent after that.

In 2012, the Supreme Court ruled that the expansion of Medicaid was an option for states, not a requirement. Thirty states have chosen to expand eligibility, and several others are negotiating with the Obama administration. But state-level brawls over Medicaid expansion have mirrored the wider political war over the law.

Gov. Bill Haslam of Tennessee, a Republican, wanted to use federal Medicaid money to extend coverage to 280,000 low-income people. His proposal failed in the spring in the legislature, under attack by conservative groups like the Koch brothers’ Americans for Prosperity, which urged voters to “stop Obamacare in Tennessee.”

No member of Congress has attacked the Affordable Care Act with more zeal than Senator John Barrasso, Republican of Wyoming. But Gov. Matt Mead of Wyoming, also a Republican, is urging state legislators to expand Medicaid to cover thousands of low-income people.

“When I came into office in 2011, I joined other states in a lawsuit challenging the Affordable Care Act, and I still don’t like it,” Mr. Mead said in an interview. “But it’s the law of land. So now I’m trying to be pragmatic, recognizing that we have about 18,000 people who could obtain coverage. We have small hospitals that are struggling. Our federal tax dollars are not headed back to Wyoming, but are paying for health care in Colorado, California and other states.”

Clinching the case for Mr. Mead is the state’s fiscal plight. Revenue is down because of a steep decline in oil and natural gas prices. With the expansion of Medicaid, he said, Wyoming would receive an infusion of federal funds, easing its budget problems.