The Senate Conservatives Fund's latest target is the GOP's leader in the Senate. 

The conservative group has launched a radio ad in Senate Minority Leader Mitch McConnellAddison (Mitch) Mitchell McConnellMulvaney aims to cement CFPB legacy by ensuring successor's confirmation Senate left in limbo by Trump tweets, House delays Political figures pay tribute to Charles Krauthammer MORE's (R-Ky.) district that scolds him for not supporting an effort that risks a government shutdown over funding ObamaCare.

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"Republicans in Congress can stop ObamaCare by refusing to fund it, but Sen. Mitch McConnell refuses to lead the fight," a narrator says in the ad.

According to The Associated Press, the ad highlights McConnell's reluctance to sign a letter in which he'd pledge his support to vote against any government funding bill that includes money for ObamaCare. 

It also alludes to McConnell campaign manager Jesse Benton's comments — from a private conversation that leaked — in which Benton said he was "holding my nose" to work with McConnell. 

"The ObamaCare bill stinks, and holding your nose won't make it any better," the narrator says.

The radio ad is backed by $50,000 and will launch Tuesday.

McConnell is facing a Tea Party challenge from Kentucky businessman Matt Bevin, and the polarizing issue of defunding ObamaCare — popular with the conservative grassroots — could become a difficult issue for him.

His campaign spokeswoman, Allison Moore, asserted in an email that it was "absurd" to suggest McConnell hadn't fought against the law.

"There is nobody in Congress who has done more to fight Obamacare since day one than Mitch McConnell and any assertion to the contrary is absurd," she said. 

Moore also suggested that McConnell is working to "unite conservatives."

"Senator McConnell is a co-sponsor of the repeal bill, a co-sponsor of the defund bill, and has said that if elected Majority Leader getting rid of Obamacare would be the first thing on his agenda.  Senator McConnell's primary goal is to unite conservatives rather than exacerbate differences to stop this train-wreck of a law and give us the best chance for success," she said.

The Senate Conservatives Fund hasn't endorsed Bevin, but its continued attacks on McConnell are an indication they're moving closer to supporting his opponent.

The group has targeted six other GOP senators who, like McConnell, have not signed the defunding ObamaCare letter: Sens. Richard BurrRichard Mauze BurrHillicon Valley: New FTC chief eyes shake up of tech regulation | Lawmakers target Google, Huawei partnership | Microsoft employees voice anger over ICE contract On The Money — Sponsored by Prudential — Senators hammers Ross on Trump tariffs | EU levies tariffs on US goods | Senate rejects Trump plan to claw back spending Senate Intel requests more testimony from Comey, McCabe MORE (N.C.), Lindsey GrahamLindsey Olin GrahamSenate panel advances three spending bills Trump says he will sign executive order to end family separations Trump backs narrow bill halting family separations: official MORE (S.C.), Lamar AlexanderAndrew (Lamar) Lamar AlexanderOn The Money — Sponsored by Prudential — Supreme Court allows states to collect sales taxes from online retailers | Judge finds consumer bureau structure unconstitutional | Banks clear Fed stress tests Supreme Court rules states can require online sellers to collect sales tax 13 GOP senators ask administration to pause separation of immigrant families MORE (Tenn.), Johnny IsaksonJohn (Johnny) Hardy IsaksonOn The Money — Sponsored by Prudential — Senators hammers Ross on Trump tariffs | EU levies tariffs on US goods | Senate rejects Trump plan to claw back spending Senators hammer Ross over Trump tariffs Overnight Finance: Senators introduce bill to curb Trump's tariff authority | McConnell calls it 'exercise in futility' | Kudlow warns WTO won't dictate policy | Mulvaney feud with consumer advocates deepens MORE (Ga.), Jeff FlakeJeffrey (Jeff) Lane FlakeMcSally takes hard line on immigration in Arizona primary Flake threatens to limit Trump court nominees: report Poll: McSally holds 14-point lead in Arizona GOP Senate primary MORE (Ariz.) and Thad CochranWilliam (Thad) Thad CochranTodd Young in talks about chairing Senate GOP campaign arm US farming cannot afford to continue to fall behind Mississippi Democrat drops Senate bid MORE (Miss.).

Many Republicans oppose the strategy of threatening to shut down the government over defunding ObamaCare, arguing the GOP will take a political hit if voters are upset by the shutdown.

--This piece was updated at 1:14 p.m. to reflect comment from McConnell's campaign.