A bipartisan group of about 40 House lawmakers are holding private talks to find a compromise to end the shutdown, Republican Congressman Reid Ribble said.
At least 15 Republicans are pressing House Speaker John Boehner to call a vote on a Senate-passed spending bill free of Obamacare-related measures. Five of them met Boehner before he and other congressional leaders met President Barack Obama at the White House.
According to two lawmakers who participated in the gathering with Boehner, the Speaker told them he wanted to shift the focus to a long-term agreement to address the nation's debt that would also avoid a clash over raising the US debt limit later this month. The lawmakers requested anonymity.
Congressman Michael Fitzpatrick, also part of the group, said Boehner "very clearly wants a long-term resolution that puts the country on a more solid economic and financial footing".
The anti-shutdown wing of the party is growing.
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 conservative "tea party" members and fought against the health care measure before it became law in 2010; 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 Congressman Frank Wolf, who is serving his 17th term in the House.
Lawmakers frustrated with the tea party's outsized influence have been working behind the scenes for weeks. Ribble has spoken out during his party's past five conference meetings to implore Boehner to redirect his party's priorities to the entitlement programmes - Medicare and Social Security - that pose the greatest threat to the nation's fiscal future.