Correction: An earlier version of this article mistakenly stated that Americans for Prosperity is backing Thom Tillis in the Republican primary for U.S. senator of North Carolina. This version has been corrected.

The fliers landed in the mailboxes of Republican voters here last week with a warning likely to unnerve many conservatives.

Thom Tillis, the Republican front-runner for a U.S. Senate seat, once called President Obama’s health-care law “a great idea,” the mailer said. The assertion echoed recent radio ads that also seem to question Tillis’s adherence to the orthodoxy of a party that has made its opposition to the Affordable Care Act a centerpiece of its midterm-election strategy.

But the warnings didn’t come from any of the seven opponents Tillis will face in Tuesday’s GOP primary, where he has been regularly attacked as not conservative enough. Instead, they were paid for by Sen. Kay Hagan, a Democrat who will face the eventual GOP nominee in November.

Hagan supports the health-care law but she is taking the unusual step of spending money on advertisements designed to appeal to Republican voters who are skeptical of the measure. The maneuver is apparently intended to undermine enthusiasm in the GOP base for the Republican who is considered her strongest potential challenger in November.

In attacking a possible rival still embroiled in a primary contest, Hagan is embracing at least a variation of a tactic other vulnerable Democratic senators have used successfully in recent elections. Senate Majority Leader Harry M. Reid (Nev.) and Sen. Claire McCaskill (Mo.), who initially faced difficult reelection odds in 2010 and 2012, respectively, found ways to reach into GOP primaries and help weaker candidates emerge as their challengers.

View Graphic Election Lab: See our current forecast for every congressional race in 2014

If Tillis, 53, does not exceed 40 percent of the vote in North Carolina’s Republican primary Tuesday, he will face a potentially costly and divisive runoff in July. Polls and interviews with strategists on both sides of the aisle suggest that Tillis is leading but could be close to hitting the 40 percent threshold. His rivals include a Baptist minister with support from former Arkansas governor Mike Huckabee and a tea party favorite who appeared at a rally Monday with Sen. Rand Paul (R-Ky.).

Tillis, speaker of North Carolina’s GOP-led House, is popular among some conservatives for helping to guide a dramatic shift to the right in Raleigh over the past four years that has spawned mass demonstrations on the left.

A vocal critic of the health-care law, Tillis said Hagan is seeking to have an impact on Tuesday’s primary. During the February radio interview that has become the basis for Hagan’s attack ads, Tillis was highly critical of the measure before concluding that it was a “great idea that can’t be paid for” — a quote he says was sarcasm that Hagan took out of context.

“They’ve meddled in the campaign. . . . They’ll be meddling in the campaign all the way to Tuesday. And if we have a runoff, they’ll be meddling again,” Tillis said as he greeted Republicans arriving to vote early at a library in Charlotte on Saturday. “They do not want to see me face off against Kay Hagan in November.”

Democratic efforts to intervene in this year’s North Carolina GOP primary may be more difficult than they were in the Reid and McCaskill races. GOP allies have flooded the airwaves in North Carolina and other competitive states this year, determined to avoid the kinds of low-caliber general-election candidates blamed for denying the party a Senate majority in 2012.

Republicans need to pick up six seats to win control of the Senate, where Democrats hold a 55 to 45 advantage and Vice President Biden has the constitutional authority to cast tie-breaking votes.

Polls suggest Hagan is one of the Senate’s most vulnerable Democrats, and any path to a GOP majority in the Senate is likely to go through North Carolina.

Groups such as the U.S. Chamber of Commerce and American Crossroads have spent millions in support of Tillis, who was elected speaker of the state House after Republicans won control of the chamber in 2010. The National Rifle Association is backing Tillis, as are former Florida governor Jeb Bush and North Carolina Gov. Pat McCrory. Mitt Romney, the party’s presidential nominee in 2012, endorsed Tillis on Monday.

Hagan, in an interview, brushed off the suggestion that she is trying to interfere with the Republican race. She said she is battling against $12 million that outside groups have spent to fight her in what is expected to be one of the most expensive Senate races.

“Every [Republican] candidate has a fringe agenda — I’ll let the primary voters Tuesday decide who my actual opponent will be,” she said.

Hagan added that her goal is to “set the record straight, after Tillis is trying to have it both ways,” running on his support for repealing the health-care law while also saying he would find a way to preserve provisions of the legislation that have proved popular.

Hagan’s campaign declined to say whether the mailers specifically targeted Republican households. Hagan spokeswoman Sadie Weiner said only that the effort was “statewide.”

Several GOP strategists said the flier was delivered to their homes in recent days, suggesting that the ad was directed at Republicans.

If forced into a runoff with either of his two top opponents, the Rev. Mark Harris or physician Greg Brannon, Tillis would spend the next two months in an expensive battle to win conservative voters. That would leave Hagan time to define herself with independent and centrist voters who are likely to decide the November election.

Potentially more worrisome for Tillis’s supporters, he would need to contend with a runoff campaign just as the North Carolina legislature returns to session later this month.

That probably would prompt a new round of protests that could help Hagan organize and excite a Democratic base that might otherwise sit out the non-presidential election.

Since Republicans took control of both chambers of the legislature in 2010 for the first time in more than 100 years, they have passed new voter identification legislation and restrictions on abortion clinics. They have cut taxes as well as education spending. Polls show that the legislature’s approval ratings are low, and Democrats think Hagan and others in the party could benefit from a renewed focus on Raleigh.

Hagan said Democratic voters, upset about the policy changes enacted in Raleigh, are “paying attention, in many cases in a way they might not have been before.”

Tillis said voters feel far more antipathy toward Washington. He said that the General Assembly does far better in polls than Congress and that he has no fears about running a campaign while leading the state House — which he said would be a good opportunity to showcase his legislative achievements.

That record, Tillis said, includes resisting efforts to expand Medicaid under the Affordable Care Act.

Still, there is no doubt that Hagan’s health-care ads have provided potent ammunition to Tillis’s Republican opponents, who have picked up on the same Tillis “great idea” quote as a reason for Republican voters to worry.

“Anything that we give [Hagan] in terms of baggage from our own lives and our own records is on the table. We should be concerned about that,” said Harris, who has built support among evangelical voters. The health-care issue “needs to be dealt with in this primary.”

At a GOP barbecue Saturday here in rural and heavily Republican Catawba County, about an hour north of Charlotte, primary voters said the Hagan ad gave them pause.

“Obamacare is not great,” said Dan Johnson, 67, a retired National Guard member from Newton who is supporting Harris. “I just think he’s one of those conservatives who’s conservative until he gets to Washington. Then he becomes part of Washington. That’s how I’ve sized him up.”