By Elise Viebeck - 07/24/13 02:57 PM EDT
The conservative Club for Growth is pushing Senate Minority Leader Mitch McConnellMitch McConnellSenate passes Puerto Rico debt relief bill Senate Appropriations speeds through spending bills The Trail 2016: When a pivot isn’t always a pivot MORE (R-Ky.) to use the threat of a government shutdown to deny funds for ObamaCare.
The group urged McConnell on Wednesday to back an effort led by Sen. Mike LeeMike LeeTwo transgender candidates win primaries Overnight Finance: Trump threatens NAFTA withdrawal | Senate poised for crucial Puerto Rico vote | Ryan calls for UK trade deal | Senate Dems block Zika funding deal Senate Democrats block Zika agreement ahead of recess MORE (R-Utah) to filibuster any government funding bill that includes money for the healthcare law.
Lee has been circulating a letter summarizing the plan. It has 15 signatures so far, according to the Club.
"We are disappointed by rumors that Senate Republican leadership is pressuring senators not to sign Lee's letter or to remove their names," Chocola added.
The push puts McConnell in a tough position.
On one hand, the GOP leader wants to ensure his party is not blamed for a government shutdown.
On the other, McConnell is facing a primary challenge from Tea Party businessman Matt Bevin, who is scheduled to officially announce his bid on Wednesday.
Groups like the Club for Growth could ultimately side with Bevin and commit serious resources to his race, posing a major threat to McConnell. The United Kentucky Tea Party endorsed Bevin ahead of his announcement Wednesday.
On Wednesday, Chocola said Club for Growth had met with Bevin and wanted to “hear more about his candidacy and the differences between him and Senator McConnell on the issues.”
“The Club’s PAC will watch Kentucky’s Senate race – as it would with any race – over the coming months to determine if our involvement is warranted,” he said in a statement.
Talk of a shutdown threat over ObamaCare has intensified as members of Senate GOP leadership lend support to the plan, including McConnell's two top deputies, John CornynJohn CornynOvernight Finance: Trump threatens NAFTA withdrawal | Senate poised for crucial Puerto Rico vote | Ryan calls for UK trade deal | Senate Dems block Zika funding deal Senate faces critical vote on Puerto Rico McConnell tees up House Puerto Rico bill MORE (Texas) and John ThuneJohn ThuneFacebook offers set of 'Values' to reassure users of neutrality Overnight Tech: Groups grade Clinton tech agenda | Facebook activates safety check in Istanbul | Another holdup for location data bill Congress prepping short-term FAA bill MORE (S.D.).
In addition to the Club for Growth, Heritage Action has endorsed the effort, urging lawmakers Tuesday only to support must-pass bills that defund ObamaCare.
This story was updated at 12:41 p.m.