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The latest Real Clear Politics average of polls shows that Mr. McConnell holds a slim lead in Kentucky, while Ms. Nunn has a slight edge over Mr. Kingston and runs neck and neck with Mr. Perdue in Georgia.

Mr. Gonzales called Ms. Nunn a “very credible candidate” but said the jury is still out on how she will perform in the general election because she is untested.

“We don’t know yet how she will hold up to increased scrutiny and criticism,” Mr. Gonzales said. “She has had the luxury of running on her own basically so far. She has been able to say whatever she says without a lot of pushback.”

Republicans say that Ms. Nunn also could get a boost from the ongoing ethics investigation into Republican Gov. Nathan Deal, who is being challenged in the fall by Democrat Jason Carter, the grandson of former President Jimmy Carter.

But they are confident that Mr. Kingston or Mr. Perdue will win in part because of blowback against President Obama and his signature domestic achievement, the Affordable Care Act.

“He is a major drag on Nunn,” said David Johnson, a Georgia-based GOP strategist. “That is the one thing going for all these Republican candidates, because you can tie the Democrats to Barack Obama and among white voters, Obama is lethal.”

The National Republican Senatorial Committee, the campaign arm of Senate Republicans, has telegraphed its general election attacks, calling Ms. Nunn the “top recruit” of Mr. Obama and Mr. Reid, and accusing her of refusing “to take a stand on issues including Obamacare.”

“Republicans’ victory tonight and momentum will finally force Nunn to come out of hiding and start answering the tough questions that she’s ducked up until now,” said Brook Hougesen, an NRSC spokesperson.

The Democratic Senatorial Campaign Committee, meanwhile, is looking to fire up its base through the “Bannock Street project,” which aims to turn out the single women, minorities and young voters that have played a big role in presidential contests but sat out midterm elections.

The DSCC also signaled that it plans to make the case that Mr. Kingston has the baggage of an unpopular incumbent lawmaker and to paint Mr. Perdue as another rich white guy who is cut from the same cloth as unsuccessful 2012 presidential nominee Mitt Romney.