Texas Gov. Rick Perry, along with a slew of other Republican governors, faces a dilemma: Do they apply for millions of dollars in federal grants by September to begin establishing state-run health insurance exchanges, or let the deadline slide, lose the federal money and risk falling into a federally run exchange?

Republican governors are unanimous in their condemnation of President Barack Obama’s health care law. But one by one, many of them are moving forward to build state exchanges, which are intended to help people not covered by large-company plans buy private health insurance at subsidized rates.

Mississippi this week received $20 million. On Monday, Idaho Republican Gov. Butch Otter announced he will accept federal money to establish an exchange. Last week, a panel appointed by Georgia’s Republican governor, Nathan Deal, gave the green light to an exchange. Those states followed Republican governors in Ohio, North Dakota, Nevada, Indiana and Wisconsin.

But Kansas’s Sam Brownback and Oklahoma’s Mary Fallin have returned tens of millions of dollars their Democratic predecessors had already received to build state exchanges.

Florida’s Rick Scott and Louisiana’s Bobby Jindal have said they will not participate, hoping either the Supreme Court or a Republican president will throw the law out before the federal government steps in.

That split has left the eyes of the health care world on governors on the fence, especially Mr. Perry, the only sitting governor running for president. With 26% of Texans uninsured, the highest share in the country, the attention there is all the more acute.

“The politics are fierce,” said Alan Weil, executive director of the National Academy for State Health Policy, a think tank tracking health law implementation.

The exchanges are the centerpiece of the 2010 health care law, designed to be government-run marketplaces where private insurers would compete to offer health plans to the uninsured and to small businesses at rates subsidized by the federal government. House Democrats and the White House wanted a single national exchange, but senators insisted they be operated by the states. The exchanges are to be running by 2014. But if a state has not made substantial progress by Jan. 1, 2013, the federal government will step in.

That threat has been enough to convince many states to move. All but Alaska accepted a $1 million federal planning grant about a year ago to look at the feasibility of state exchanges. Sixteen states and the District of Columbia have received much larger “establishment” grants to begin building the insurance marketplaces, including three—Indiana, Mississippi and Nevada—headed by Republicans. Next month is the deadline for the first round of such grants.

Other Republican governors who have neither received an establishment grant nor said they wouldn’t apply include Chris Christie of New Jersey, Terry Branstad of Iowa, Nikki Haley of South Carolina, Susana Martinez of New Mexico and Tom Corbett of Pennsylvania.

Mr. Perry has sent mixed signals. With a veto threat, he killed a Republican-led effort in the Texas legislature to establish a Texas exchange, said state Rep. John Zerwas, the Republican author of the bill. But he also signaled he could move forward on his own if a multi-state legal challenge or repeal efforts falter.

“The governor has made it clear that he believes Texans should be in charge of our health care programs, and that he would prefer local or state solutions to Texas’s health care challenges, rather than one-size-fits-all bureaucracy from the federal government,” said Perry spokeswoman Lucy Nashed. “We continue to be hopeful that the court cases declaring Obamacare unconstitutional are upheld.”

For Mr. Perry, the dilemma is acute. Grassroots conservatives, the mainstay of his presidential support, see any move to comply with the federal law as acquiescing to it, said Mr. Zerwas, a medical doctor and health-care law opponent. But by not moving forward, states risk giving the federal health bureaucracy more power, not less.

“I don’t want to look like someone who’s defending Obamacare, so I’ve made this about the sovereignty of Texas,” he said. “But exchanges have become synonymous with Obamacare, whether we like it or not.”

Complicating the Perry decision is the fact that Texas’s biennial legislature will not meet again until 2013, after the federal deadline. Mr. Zerwas said Mr. Perry personally assured him he could implement the exchange through executive action, if needed. But that is not clear.

“We’re directed by state law. There would need to be some action taken in the legislature for there to be a state-run health exchange,” said John Greeley, a spokesman for the Texas Department of Insurance.

Sandy Praeger, the Republican insurance commissioner in Kansas, said her party shouldn’t put its faith in the law’s repeal. The federal exchange-takeover deadline is Jan. 1, 2013. A Republican president would not take office until Jan. 20, 2013. Repeal legislation could be filibustered in the Senate, and components of the law already in effect—among them, restrictions on annual coverage caps and allowing young people to stay on their parents’ insurance—are broadly popular.

“The system we have in this country needs fixing, and while the health care law is not perfect, it’s a step toward getting people covered,” Ms. Praeger said. “We need to move forward.”

But her own governor disagrees.

“There is much uncertainty surrounding the ability of the federal government to meet its already budgeted future spending obligations,” Mr. Brownback said in a statement this month, rejecting a $31.5 million federal grant to build the Kansas exchange. “Every state should be preparing for fewer federal resources, not more.”

Write to Jonathan Weisman at jonathan.weisman@wsj.com