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Former Gov. Jeb Bush of Florida and Gov. Mary Fallin of Oklahoma participated in a discussion with students at KIPP Academy in Oklahoma City in April.CreditJim Beckel/The Oklahoman, via Associated Press

WASHINGTON — The health care law may be Republicans’ favorite weapon against Democrats this year, but there is another issue roiling their party and shaping the establishment-versus-grass-roots divide ahead of the 2016 presidential primaries: the Common Core.

A once little-known set of national educational standards introduced in 44 states and the District of Columbia with the overwhelming support of Republican governors, the Common Core has incited intense resistance on the right and prompted some in the party to reverse field and join colleagues who believe it will lead to a federal takeover of schools.

Conservatives denounce it as “Obamacore,” in what has become a surefire applause line for potential presidential hopefuls. Other Republicans are facing opprobrium from their own party for not doing more to stop it. At a recent Republican women’s club luncheon in North Carolina, a member went from table to table distributing literature that called the program part of “the silent erosion of our civil liberties.”

The learning benchmarks, intended to raise students’ proficiency in math and English, were adopted as part of a 2010 effort by the National Governors Association and theCouncil of Chief State School Officers to bolster the country’s competitiveness. Some conservatives, in an echo of their criticism of the health care law, say the standards are an overreach by the federal government.

Yet there is an important distinction: Unlike the health care law, the Common Core retains bipartisan support and has the backing of powerful elements of the business community.

Its most outspoken Republican defender, former Gov. Jeb Bush of Florida, is also the most talked-about potential presidential candidate among mainstream party leaders and donors. Mr. Bush has called out some Republicans who have switched positions, drawing what will be a dividing line in the campaign if he or other defenders of the Common Core choose to run. He is joined by Gov. Chris Christie of New Jersey, but theirs is becoming a small club.

“I’m a big fan of Jeb Bush; I think he’s an important leader on many issues,” said Senator Ted Cruz, Republican of Texas. “But on the question of Common Core, I emphatically do not agree with Common Core.” His opinion of the program is shared by two Senate colleagues and possible 2016 rivals for the presidential nomination, Senators Rand Paul of Kentucky andMarco Rubio of Florida.

Mr. Cruz’s view also aligns with that of several Republican governors contemplating presidential runs. Gov. Mike Pence of Indiana signed legislation last month that made his state the first to opt out of the Common Core after having adopted it. Gov. Scott Walker of Wisconsin said he wanted his state to establish its own educational goals. And Gov. Bobby Jindal of Louisiana suggested that he might use executive authority to go around the State Legislature if lawmakers did not withdraw from the group of states developing the standardized test associated with the Common Core.

Mr. Jindal’s position, a reversal for him, shows how quickly conservative opposition has grown. He recently announced his support for a bill that would remove Louisiana from the Common Core, on the same day the U.S. Chamber of Commerce Foundation, which supports the program, released a video featuring his earlier endorsement of it.

The Republican revolt against the Common Core can be traced to President Obama’s embrace of it, particularly his linking the adoption of similar standards to states’ eligibility for federal education grants and to waivers from No Child Left Behind, the national education law enacted by PresidentGeorge W. Bush.

It underlines the ascendance of a brand of conservatism notably different from that of the most recent President Bush. Less than 15 years after No Child Left Behind passed with just 34 House Republicans opposed to it, the conservative center of gravity is shifting toward a state-centric approach to education.

“When I arrived on Capitol Hill in 2001, not only was the Republican administration not devolving power to the states, the No. 1 priority of the administration was a massive expansion of the federal Department of Education,” recalled Mr. Pence, who, as a congressman, opposed No Child Left Behind.

The opposition to the Common Core also captures another shift since the Bush administration: While long contemptuous of an expanding federal government, some Republican activists are growing wary of big business, too, including figures like Bill Gates, the billionaire Microsoft founder whose foundation supported the development of the standards.

“There is a legitimate concern about large institutions, be they government or others, who haven’t really delivered the America everybody thought we were on our way to,” acknowledged John R. McKernan Jr., a former Maine governor who leads the U.S. Chamber of Commerce Foundation. But, he said, that fear is “totally misplaced” when it comes to the Common Core.

But would-be presidential candidates are paying more heed to the conservative activists holding packed meetings in their states and flooding them with emails. “The Republican Party is getting more and more responsive to the grass roots, and that is a very healthy thing for the party and the country,” Mr. Cruz said.

Jeb Bush said the pivot seemed more like pandering. In remarks this month during an event at his father’s presidential library, he affirmed his support for the Common Core. “I guess I’ve been out of office for a while, so the idea that something that I support — because people are opposed to it means that I have to stop supporting it if there’s not any reason based on fact to do that?” he said. “I just don’t feel compelled to run for cover when I think this is the right thing to do for our country.”

With a knowing grin, he added, “Others that supported the standards all the sudden now are opposed to it.”

Some other former Republican governors who pushed the adoption of the Common Core agree with Mr. Bush. “There is a great deal of paranoia in the country today,” said Sonny Perdue, a former governor of Georgia, who was also instrumental in creating the program. “It’s the two P’s, polarization and paranoia.”

Supporters of the Common Core, which outlines skills that students in each grade should master but leaves actual decisions about curriculum to states and districts, say that it was not created by the federal government and that it was up to the states to decide whether to adopt the standards.

But opponents say Mr. Obama’s attempt to reward states that adopt the standards with grants and waivers amounts to a backdoor grab for federal control over what is taught in schools.

“Standards inevitably influence the curricula being taught to meet those standards,” Mr. Cruz said.

It is not just conservatives who have turned against the Common Core: The leaders of major teachers unions are also pushing back because of the new, more difficult tests aligned to the standards that are being used to evaluate both students and teachers.

“You have this unlikely marriage of folks on the far right who are convinced this is part of a federal takeover of local education, who have joined hands with folks on the left associated with teachers unions who are trying to sever any connection between test results and teacher evaluation,” said Gov. Bill Haslam of Tennessee, a Republican who supports the Common Core.

But it is on the right that the anger is growing. A recent forum on the Common Core in Columbus, Ohio, drew 500 people, most of them concerned parents, said Jane Robbins, a senior fellow at the American Principles Project, a conservative group opposed to the program. Such meetings reflect discontent that is bubbling up at the local level, where some county Republican committees are moving to punish legislators who do not oppose the standards.

“I think the establishment in the party has been slow to recognize how big this is,” Ms. Robbins said.