By Elise Viebeck - 07/24/13 02:57 PM EDT
The conservative Club for Growth is pushing Senate Minority Leader Mitch McConnellMitch McConnellPeter Thiel does not make the GOP pro-gay Reid: Trump is a 'hateful con man' McAuliffe: Clinton won't move TPP without changes 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 LeeObama signs opioid bill Thiel said to explain support for Trump in convention speech Convention erupts at Cruz snub 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 CornynGOP senators to donors: Stick with us regardless of Trump Hopes dim for mental health deal Overnight Finance: Senate punts on Zika funding | House panel clears final spending bill | Biz groups press Treasury on tax rules | Obama trade rep confident Pacific deal passes this year MORE (Texas) and John ThuneJohn ThuneTim Kaine backs call to boost funding for Israeli missile defense FCC chief pushes phone companies to offer free robocall blocking How the new aviation law will affect your travel 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.