New Congress live updates: What's next after Kevin McCarthy's speakership win

During a conference call this morning with the Republican conference, McCarthy told members that there's not officially a deal to vote him speaker, but said he believes they're "in a good position" and working in "good faith."

"I'm not telling you we have an agreement," McCarthy said on the call, according to multiple sources. "We're in a good position and having meetings."

McCarthy ally, Patrick McHenry, said on the call that no gavels for committee chairmanships have been promised during their talks.

McCarthy, again, said on this call that he has agreed to offer a key concession: making it even easier to remove him as speaker if he is elected. So, a single member, from either party, can trigger an up-or-down simple majority vote on whether to remove the speaker.

But still, after failing 11 times, the math still doesn't appear to be in McCarthy's favor. And there could be votes over the weekend.

Rep. Brian Fitzpatrick, a pro-McCarthy ally, told ABC News that the next phase of working on the handful of hard no's will involve more isolated discussions with those who have concerns beyond just personal issues.

"It's not a monolithic group. Each one of them have different concerns. Some, it's motion to vacate, some, it's guardrails around the budget. But when you shrink that group down, you're able to have one-on-one conversations as to what does it take for them to get to yes, if anything," Fitzpatrick said.

-ABC News Congressional Correspondent Rachel Scott, Katherine Faulders and Will Steakin

The vast majority of the Republicans blocking McCarthy's speaker bid have given life to the so-called 'big lie' in the two years since the Capitol attack.

Of the 15 incumbents in the flank rejecting McCarthy, 14 challenged the results of the 2020 election two years ago.

Just two of the 20, which includes five freshmen, acknowledge the legitimacy of the 2020 election: Chip Roy of Texas and Rep.-elect Josh Brecheen of Oklahoma.

The small but significant group appears on the verge now of taking the reins of power in Washington by using their leverage against McCarthy, who himself has a tortured relationship with Jan. 6 and the 2020 election.

McCarthy was one of 147 House Republicans to reject its certification ahead of a floor speech deeming Trump "bears responsibility" for the "mob attack."

-ABC News' Trish Turner

With all other House business put on hold, Republican Rep. Don Bacon of Nebraska has continued to voice frustration with the small but significant flank of his party refusing to support McCarthy's speaker bid.

"Our constituents call me and they want help with passports, visas, disability claims with the VA, and all that's on hold -- and that's because of these 20 people who are holding us hostage," Bacon told ABC News Congressional Correspondent Rachel Scott.

Bacon, who serves on the House Committee on Agriculture and House Committee on Armed Services, said he was also forced to cancel a classified briefing this week because he lost his security clearance.

Kevin McCarthy and his loyalists have been forced to make waves of concessions that will likely make the job of any GOP speaker harder. Behind arcane arguments about such things as "legislative germaneness" and a "motion to vacate" are real pending rules changes that could tie up the House floor indefinitely -- and limit the latitude any speaker has to make unpopular decisions.

That could matter deeply in the case of a true crisis, where speed of action and a House speaker's input could be critical. It will also almost certainly matter when it comes to basic matters of funding the federal government and making good on debts already incurred, which forces Congress to periodically raise the nation's borrowing limit.

Legislation in those areas almost always requires a modicum of bipartisanship -- particularly so long as the House is controlled by a different party than the Senate and the White House, as it is now. That means cutting deals that many in a speaker's own party may not agree with, without having to worry about whether that risks a no-confidence vote that could cost the gavel.

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-ABC News' Political Director Rick Klein