ATLANTA — Florida Gov. Rick Scott, who made a fortune as a health care executive, long opposed President Barack Obama’s remake of the health insurance market. After the Democratic president won re-election, the Republican governor softened his tone. He said he wanted to “have a conversation” with the administration about implementing the 2010 law. With a federal deadline approaching, he also said while Florida won’t set up the exchange for individuals to buy private insurance policies, the feds can do it.

In New Jersey, Gov. Chris Christie held his cards before saying he won’t set up his own exchange, but he’s avoided absolute language and says he could change his mind. He’s also leaving his options open to accept federal money to expand Medicaid insurance for people who aren’t covered. The caveat, Christie says, is whether Health Secretary Kathleen Sebelius can “answer my questions” about its operations and expense.

Both Republican governors face re-election in states that Obama won twice, Christie in 2013 and Scott in 2014. And both will encounter well-financed Democrats.

Their apparent struggles on the issue, along with other postures by their GOP colleagues elsewhere, suggest political uncertainty for Republicans as the Affordable Care Act starts to go into effect two years after clearing Congress without a single Republican vote. The risks also are acute for governors in Democratic-leaning or swing-voting states or who know their records will be parsed should they seek the presidency in 2016 or beyond.

“It’s a tough call for many Republican governors who want to do the best thing for their state but don’t want to be seen as advancing an overhaul that many Republicans continue to detest,” said Whit Ayers, a consultant in Virginia whose clients include Gov. Bill Haslam of Tennessee, a Republican who didn’t announce his rejection of a state exchange until days before Sebelius’s Dec. 14 deadline.

Indeed, cracks keep growing in the near-unanimous Republican rejection of Obama’s health care law that characterized the GOP’s political messaging for the last two years. Five GOP-led states — Idaho, Mississippi, Nevada, New Mexico, and Utah — are pressing ahead with state insurance exchanges. Ongoing monitoring by The Associated Press shows that another five Republican-led states are pursuing or seriously a partnership with Washington to help run the new markets.

Democrats, meanwhile, hope to use the law and Republican inflexibility to their advantage, betting that more Americans will embrace the law once it expands coverage. The calculus for voters, Democrats assume, will become more about the policy and less about a polarizing president.

“It shouldn’t be complicated at all,” said John Anzalone, an Obama pollster who assists Democrats in federal races across the country. Anzalone said Republicans could use their own states-rights argument to justify running exchanges. Instead, he said, “They are blinded by Obama-hatred rather than seeing what’s good for their citizens.”

Governors can set up their own exchanges, partner with Sebelius’ agency or let the federal government do it. The exchanges are set to open Jan. 1, 2014, allowing individuals and businesses to shop online for individual policies from private insurers. Low- and middle-income individuals will get federal premium subsidies calculated on a sliding income scale. Eighteen states plus Washington, DC, most led by Democrats, have committed to opening their own exchanges.

The law also calls for raising the income threshold for Medicaid eligibility to cover people making up to 138 percent of the federal poverty line, or about $15,400 a year for an individual. That could add more than 10  million people, most of them childless adults, to the joint state-federal insurance program for low-income and disabled Americans. Together, the exchanges and the Medicaid expansion are expected to reduce the number of uninsured by about 30 million people within the next decade.

A  Supreme Court ruling last summer made the Medicaid expansion voluntary, rather than mandatory for states. At least eight governors, all of them Republicans, have already said they have no plans to expand Medicaid.

The complexity is obvious.

National exit polls from last month’s election showed that 49 percent of voters wanted some or all of Obama’s signature legislative achievement rolled back. Among self-identified independents, that number was 58 percent. Among Republicans, it spiked to 81 percent. When asked about the role of government, half of respondents said the notion that government is doing too much fits their views more closely than the idea that government should do more.

Before the election, a national AP-GfK poll suggested that 63 percent of respondents preferred their states to run insurance exchanges, almost double the 32 percent who wanted the federal government to take that role. And the same electorate that tilts toward repealing some or all of the new law clearly re-elected its champion.

That’s not the most important consideration for governors who face re-election in Republican states. Georgia’s Nathan Deal and Alabama’s Robert Bentley, who also face 2014 campaigns, initially set up advisory commissions to consider how to carry out the health care law, but they’ve since jumped ship. But, unlike others, Deal and Bentley aren’t eyeing national office.

Three Republicans who are viewed as potential national candidates — Rick Perry of Texas, Nikki Haley of South Carolina and Bobby Jindal of Louisiana — were full-throated opponents. Jindal, the only one of the three who is term-limited, is the incoming chairman of the Republican Governors Association. In that role, he has co-signed more conciliatory letters to Sebelius asking questions to flesh out how the designs might work.

Republican governors also are feeling quiet pressure from hospitals and other providers.

Deal, the Georgia governor, offers the typical argument for saying no: “We can’t afford it.” But the law envisions the new Medicaid coverage more or less as a replacement of an existing financing situation that pays hospitals to treat the uninsured. The law contemplates cuts in that program, which already requires state seed money. The idea was that expanding Medicaid coverage would reduce “uncompensated care” costs.

“Some of those cuts were made with the expectation that Medicaid would be expanded and that hospitals would be paid for portions of business that we are not being paid for now,” said Don Dalton of the North Carolina Hospital Association.

Dalton’s Governor-elect, Republican Pat McCrory, said as a candidate that he opposed Medicaid expansion. Dalton said his industry is leaning on McCrory and legislative leaders, though he commended “their deliberate approach.” Similar efforts are underway in South Carolina, Georgia, Missouri and elsewhere.

For Democrats, Anzalone said the framing will be simpler: “You don’t want to take a 9-to-1 match? That’s a pretty easy investment. These governors who aren’t expanding Medicaid, they’re basically giving taxpayer money to the states that do.”