"The only way we're going to get Boehner and Cantor to change course is if we can bring things to a halt," said the source, who asked to speak anonymously while talks continue.
A perilous strategy
But it could be risky for these House Republicans to take a stand against the tea party faction of the GOP.
At the weekly lunch of the Republican Study Committee, a group of fiscal conservatives, the rumblings of the moderate GOP members came up. Some in the room said they should "go after" those fellow Republicans and put pressure on them to fall in line, according to a GOP source familiar with the discussions.
But another Republican congressional source in the meeting said the message was softer.
Members of the committee were encouraged to have "one-one-one converstations" with moderates to convince them to stick with the current GOP leadership strategy.
Nunes told reporters he expected the shutdown to go through the weekend and possibly through mid-October when Congress needs to raise the debt ceiling. He doesn't think the current House Republican plan, which he repeatedly said is being dictated by Cruz, is helping the GOP cause of defunding or delaying Obamacare.
But he said he will vote for the smaller spending bills out of loyalty to Boehner, even as he criticized the group behind Cruz as "lemmings."
"I'm going to continue to support our leadership. Even if we have entered the valley of death, when you enter the valley of death you have to keep running and the whole team has to stick together," a frustrated Nunes told reporters outside the House floor.
King acknowledged the effort to get more Republicans to push for a clean spending bill could take some time and probably wouldn't result in a new strategy until "the tea party has had enough."