Originally Published: 05 JAN 23 05:06 ET

Updated: 05 JAN 23 09:00 ET

By Clare Foran and Melanie Zanona, CNN

(CNN) -- Kevin McCarthy faces growing pressure to end the impasse over his imperiled speakership bid after two consecutive days of failed votes.

But even after proposing major concessions to his hardline opponents late Wednesday, it remains unclear if the California Republican will be able to lock in the 218 votes he needs to win the gavel, and patience is wearing thin among lawmakers as the fight drags on.

The House is set to reconvene on Thursday at 12 p.m. ET. It's unclear if a seventh vote on McCarthy's speakership will happen then or if Republicans will move to adjourn. McCarthy is leery of having additional votes that show 20 members are opposed to him and he wants to demonstrate some forward momentum, a source familiar with the matter tells CNN.

Republican holdouts were meeting Thursday morning, said Rep. Warren Davidson, who supports McCarthy.

"I think after that, they'll talk to Leader McCarthy and hopefully close up a deal," the Ohio Republican told CNN's Kaitlan Collins on "CNN This Morning," though he conceded that McCarthy "may never make it to 218."

McCarthy, however, struck an optimistic tone when he arrived on Capitol Hill.

"I think we're making progress," he said.

McCarthy agrees to additional concessions

There are some early indications that negotiations have made some headway as McCarthy and his allies attempt to chip away at opposition from a bloc of conservatives.

In a series of new concessions first reported by CNN Wednesday night, McCarthy agreed to propose a rules change that would allow just one member to call for a vote to oust a sitting speaker, according to two sources familiar with the matter. McCarthy had initially proposed a five-member threshold, down from current conference rules that require half of the GOP to call for such a vote.

He also agreed to allow for more members of the Freedom Caucus to serve on the powerful House Rules Committee, which dictates how and whether bills come to the floor, and to vote on a handful of bills that are priorities for the holdouts, including proposing term limits on members and a border security plan.

Republican sources say that even if McCarthy's offers are accepted, it would still not get him the 218 votes he needs to be speaker. While these concessions could attract some new support, other opponents have raised different concerns that have yet to be fully addressed.

McCarthy said Wednesday evening that there was no deal yet to end the stalemate, but that there has been progress. "I think it's probably best that people work through some more," McCarthy said after the House had adjourned.

McCarthy has already made a number of concessions to his opponents, though so far his efforts have not been enough.

But sources said the talks Wednesday between McCarthy allies and holdouts have been the most productive and serious ones to date. And in one sign of a breakthrough, a McCarthy-aligned super PAC agreed to not play in open Republican primaries in safe seats -- one of the big demands that conservatives had asked for but that McCarthy had resisted until this point.

Texas Rep. Chip Roy, one of the conservatives who has voted against McCarthy's speakership bid, told GOP leaders that he thinks he can get 10 holdouts to come along if ongoing negotiations pan out, according to GOP sources familiar with the internal discussions, and that there are additional detractors who may be willing to vote "present."

Still, even if these negotiations prove successful and 10 lawmakers do flip to McCarthy's column -- which is far from certain -- that won't get McCarthy to the 218 votes needed to win the speakership, so he would still have more work to do.

McCarthy also met separately Wednesday with the freshmen members-elect who voted against him, sources told CNN.

During the meeting, McCarthy reiterated some of the things he has already promised and went into greater detail about those concessions.

McCathy's direct outreach to the freshmen-elect offers another window into his strategy for winning over the holdouts.

Incoming House Majority Whip Tom Emmer commented that the negotiations have been "very, very constructive."

"There were a whole bunch of members that were involved in this, and there are some folks now that are sitting down and talking about that discussion to see where they want to go with it next," the Minnesota Republican said.

One moderate Republican told CNN Thursday morning that they aren't happy about the concessions, though they are willing to have "discussions" about them.

The fear is that lowering the threshold for a vote to oust the speaker to one member will make governing on items like the debt limit and funding almost impossible.

"I don't like the rules but am willing to hear discussions. I think they're a mistake for the conference. These handful of folks want a weak speaker with a four-vote majority. The public will not like what they see of the GOP, I fear," the member said.

House in chaos over speaker fight

The fight over the speakership, which began Tuesday on the first day of the 118th Congress, has thrown the new House GOP majority into chaos and undercut the party's agenda.

As the fight has stretched out, the situation has grown increasingly dire for McCarthy's political future as even some of his Republican allies have begun to fear that the House GOP leader may not be able to pull off his gamble for speaker if the fight goes much longer.

McCarthy has so far come up short in six rounds of voting. The final GOP tally for the sixth vote, which took place on Wednesday, was 201 for McCarthy, 20 for Florida Rep. Byron Donalds of Florida and one "present" vote.

The House will remain paralyzed until this standoff is resolved. This is the first time an election for speaker has gone to multiple ballots since 1923.

To be elected speaker, a candidate needs to win a majority of members who vote for a specific person on the House floor. That amounts to 218 votes if no member skips the vote or votes "present."

House Republicans won 222 seats in the new Congress, so for McCarthy to reach 218, he can only afford to lose four GOP votes.