PHOENIX — Like many Republican governors, Jan Brewer of Arizona is a stinging critic of President Obama’s health care law. When the Supreme Court upheld it in June, she called the ruling “an overreaching and unaffordable assault on states’ rights and individual liberty.”

Yet the Brewer administration is quietly designing an insurance exchange — one of the most essential and controversial requirements of the law. Officials in a handful of other Republican-led states say they are also working to have a framework ready by Nov. 16, the deadline for states to commit to running an exchange or leave it to the federal government to run it for them. That is just 10 days after Election Day, which is likely to decide the future of the law.

Given that the health care overhaul remains a lightning rod — just last week, Oklahoma revised a lawsuit against it — even the most tentative discussions about carrying it out in Republican states tend to take place behind closed doors or “underground,” as the leader of a health care advocacy group in the South put it.

In Mississippi, Mike Chaney, the insurance commissioner, who is laying the groundwork for a state-based exchange there, recently learned the difficulties of moving forward in anything but the utmost secrecy. At a luncheon this summer he found himself facing down an opponent of the law in a confrontation that is now circulating on YouTube.