WASHINGTON — After two years of political battles and a Supreme Court case, many if not most states are expected to tell the federal government Friday if they’re willing carry out a key part of President Barack Obama’s health care overhaul.

At issue is the creation of new health insurance markets, where millions of middle-class households and small businesses will shop for private coverage. The so-called exchanges will open for business Jan. 1, 2014, and most of their customers will be eligible for government subsidies to help pay premiums. The exchanges will also steer low-income people into expanded Medicaid programs, if states choose to broaden their safety net coverage.

Thursday evening, the Obama administration responded to a request for more time from Republican governors by granting states a month’s extension, until Dec. 14.  

Ahead of the original deadline, a check by The Associated Press found that 21 states plus the District of Columbia, have already indicated they want to become involved, either by building and running their own exchanges or partnering with Washington. The 16 that want to build their own exchanges, plus the District of Columbia, face a Jan. 1 deadline for the federal government to approve their plans.

This group of 16 includes mainly Democratic-led states such as California and New York, but also some Republican-led ones such as Mississippi and New Mexico.

Five other states have signaled they want to partner with the federal government. Those states would handle consumer issues and oversight of health plans in the exchanges, while the feds do the heavy lifting by enrolling individuals for coverage and determining who’s eligible for government assistance. Among these states are Arkansas and North Carolina.

The number of partnership states could grow significantly, since the Obama administration has given states until next February to decide on that option. As of Thursday, 16 states indicated that they were weighing their options and have not made a final decision.

Among those, Ohio and Tennessee were considering the partnership route. And in Florida, Republican Gov. Rick Scott is now saying he wants to find a way to work with the federal government after years of steadfastly opposing Obama’s overhaul.

Finally, 13 states have indicated they will default to the federal government, allowing Washington to set up and run their exchanges. The health care law provided that the feds would run exchanges in states that were not ready or willing to do so. In this group are states whose Republican governors have staunchly opposed the law, including Texas, Louisiana and South Carolina.

Obama’s election victory guaranteed the survival of his health care law, which is eventually expected to provide coverage to more than 30 million people through the exchanges and expanded Medicaid programs. It was the final hurdle, after the Supreme Court upheld a legal challenge from 26 states. In the aftermath of the election, some Republican state leaders say it’s time to accept the law.

“I don’t like it; I would not vote for it; I think it needs to be repealed. But it is the law,” said Mississippi Insurance Commissioner Mike Chaney, after announcing that his state wants to set up its own exchange. “If you default to the federal government, you forever give the keys to the state’s health insurance market to the federal government.”

Traditionally, states have regulated the private health insurance market.

But other Republican-led states say they don’t have enough information to make a decision at this point and are clamoring for the Obama administration to release major regulations that have been bottled up for months.

“States are struggling with many unanswered questions and are not able to make comprehensive far-reaching decisions prudently,” Govs. Bob McDonnell of Virginia and Bobby Jindal of Louisiana wrote Obama earlier this week. They asked for a meeting with the president, as well as a postponement of the original Nov. 16 deadline.

Some of their main concerns are hidden costs of operating the exchanges and the sheer bureaucratic complexity of the new system. The Obama administration has steadfastly maintained it will not postpone the Jan. 1, 2014, launch date for the law’s coverage expansion. Open enrollment for exchange plans will begin even sooner, Oct. 1, 2013.

Policy experts in Washington are noticing the shift.

“I think it’s a very practical decision for states now,” said Alan Weil, executive director of the nonpartisan National Academy for State Health Policy. “We are going to have a significant number of states running their own exchanges, a significant number where the federal government is running the exchange and a significant number of partnerships. The bottom line is, we are going to have to figure out how to make all three models work.”

Although the public remains divided about the health care law, the idea of states running the new insurance markets is popular, especially with Republicans and political independents. A recent AP poll found that 63 percent of Americans would prefer states to run the exchanges, with 32 percent favoring federal control.

The breakdown among Republicans was 81 percent to 17 percent in favor of state control, while independents lined up 65-28 for states taking the lead. Democrats were almost evenly divided, with a slim majority favoring state control.