Michael Steel, the speaker's spokesman, didn't comment on specific meetings with members, saying his boss "constantly listens to members from every part of our conference."
The anti-shutdown wing of the party is growing, including Virginia Reps. Frank Wolf and Scott Rigell.
Their judgment is that the party is being hurt by pursuing a strategy first championed by Republican Sen. Ted Cruz of Texas, a freshman lawmaker aligned with the small-government tea party movement who attacked Obamacare on Sept. 24 in a 21-hour speech on the Senate floor.
The Republicans pushing Boehner to work out an end to the government shutdown also oppose Obamacare. Many of them have served in Congress longer than most tea party members and fought against the health-care measure before it became law in 2010 and since then have sought to roll it back. Now they say that in the face of united opposition to such efforts by the Senate's Democratic majority, Republicans must move on.
"It's clear that you aren't going to be successful," said Wolf, who is serving his 17th term in the House. "You can argue about Obamacare, it's a bad, bad bill. But you fight that in a different way, you don't shut the government down when we are at war," said Wolf, whose Northern Virginia district is home to many of the idled federal workers and contractors.
Ribble said he is among about 40 House Republicans and Democrats who, during a meeting Wednesday, discussed a compromise involving repealing a medical device tax as part of a stopgap spending bill to reopen the government. Grimm confirmed this was among the options pitched to Boehner.
Repealing the 2.3 percent tax on medical equipment mandated under Obamacare won the support of 34 Senate Democrats and every Republican in March as part of a budget blueprint.