The conservative Club for Growth is pushing Senate Minority Leader Mitch McConnellMitch McConnellThough flawed, complex Medicaid block grants have fighting chance Sanders: 'If you don't have the guts to face your constituents,' you shouldn't be in Congress McConnell: Trump's speech should be 'tweet free' 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 LeeLessons from the godfather of regulatory budgeting Congress must reform civil asset forfeiture laws A guide to the committees: Senate 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 CornynCornyn: Border wall 'makes absolutely no sense' in some areas Ryan on border: ‘We will get this done’ Ryan tours Mexican border on horseback MORE (Texas) and John ThuneJohn ThuneYahoo reveals new details about security Conquering Trump returns to conservative summit Low-income consumer broadband credits mean competitiveness, choice and compassion 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.