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 McConnellKey GOP senators appear cool to Kavanaugh accuser's demand Trump hints at new executive action on immigration, wants filibuster-proof Senate majority The Hill's Morning Report — Sponsored by United Against Nuclear Iran — The Hill interviews President Trump 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 BurrTrump assures storm victims in Carolinas: 'We will be there 100 percent' Overnight Energy: Trump rolls back methane pollution rule | EPA watchdog to step down | China puts tariffs on US gas Graham: Mueller is going to be allowed to finish investigation MORE (N.C.), Lindsey GrahamLindsey Olin Graham GOP senator: Kavanaugh accuser 'moving the goalposts' Collins: Kavanaugh accuser should 'reconsider,' testify on Monday Grassley willing to send staff to California to speak with Kavanaugh accuser MORE (S.C.), Lamar AlexanderAndrew (Lamar) Lamar AlexanderOvernight Health Care: Senators target surprise medical bills | Group looks to allow Medicaid funds for substance abuse programs | FDA launches anti-vaping campaign for teens The Hill's Morning Report — Sponsored by United Against Nuclear Iran — Kavanaugh confirmation in sudden turmoil Cruz gets help from Senate GOP in face of serious challenge from O’Rourke MORE (Tenn.), Johnny IsaksonJohn (Johnny) Hardy IsaksonTrump blasts Tester at Montana rally: 'He loves the swamp' Renaming Senate office building after McCain sparks GOP backlash GOP senator warns Trump: Anyone who trash-talks McCain 'deserves a whipping' MORE (Ga.), Jeff FlakeJeffrey (Jeff) Lane FlakeThe Memo: Tide turns on Kavanaugh Key GOP senators appear cool to Kavanaugh accuser's demand Gillibrand: Kavanaugh accuser shouldn't participate in 'sham' hearing MORE (Ariz.) and Thad CochranWilliam (Thad) Thad CochranGOP Senate candidate to African Americans: Stop begging for 'government scraps' Trump endorses Hyde-Smith in Mississippi Senate race GOP Senate candidate doubles down on Robert E. Lee despite Twitter poll 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.