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 McConnellMitch McConnellGOP senator: Don't expect Trump to 'have your back' on healthcare vote Senate Dems step up protests ahead of ObamaCare repeal vote Johnson becomes fourth GOP senator unwilling to proceed on healthcare bill 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 BurrSenate intel panel to hold hearing on Russian meddling in Europe The Hill's Whip List: Senate ObamaCare repeal bill Overnight Tech: Uber CEO resigns | Trump's Iowa tech trip | Dems push Sessions to block AT&T-Time Warner deal | Lawmakers warned on threat to election systems | MORE (N.C.), Lindsey GrahamLindsey GrahamGOP senator: Don't expect Trump to 'have your back' on healthcare vote Five takeaways from the CBO score on Senate ObamaCare bill New CBO analysis imperils GOP ObamaCare repeal MORE (S.C.), Lamar AlexanderLamar AlexanderThe Hill's Whip List: Senate ObamaCare repeal bill Trump administration pays June ObamaCare subsidies to insurers Republicans and the lost promise of local control in education MORE (Tenn.), Johnny IsaksonJohnny IsaksonThe Hill's Whip List: Senate ObamaCare repeal bill Georgia special election runoff: live coverage House approves VA bill, sending it to Trump MORE (Ga.), Jeff FlakeJeff FlakeSenate should seek to retain its 'blue slip' tradition for judicial nominees Progressives target Heller and Flake on Senate GOP bill The Hill's Whip List: Senate ObamaCare repeal bill MORE (Ariz.) and Thad CochranThad CochranOvernight Finance: CBO finds 22M more uninsured under Senate health bill | GOP agrees ObamaCare taxes must go | Supreme Court to look at Dodd-Frank whistleblower protections | More tax reform hearings | Green light for partial travel ban | Senate Dems set principles for potential budget negotiation Defense hawks gird for budget brawl 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.