Here’s the latest on the speaker vacancy.

House Republicans have elected two nominees for the job since a hard-right clutch of lawmakers toppled Speaker Kevin McCarthy. But both men — Representatives Steve Scalise of Louisiana, the No. 2 Republican, and Jim Jordan of Ohio — were forced to withdraw their candidacies after it became clear that neither could muster the 217 votes needed to win the position.

House Republicans cast aside Mr. Jordan as their nominee for speaker on Friday in a secret-ballot vote, essentially moving to begin the search for a new leader all over again. In the days since, a flood of lawmakers have thrown their hats in the ring for the top leadership post.

Here’s a look at what comes next:

House Republicans met on Monday to hold a candidate forum for aspiring speaker nominees to present their visions for the conference.

They are holding an internal election for a new nominee on Tuesday — and if they elect one, Republicans could go to the House floor for a vote later that day.

But because nine Republicans are running for the job, that internal election may take longer than usual. Conference rules mandate that the party nominee must capture a simple majority of votes. If no one captures a majority on the first ballot, the candidate who received the fewest votes will be kicked off the second ballot and lawmakers will vote again. That process will continue until there is a nominee.

Looming over the process is the work of the House that has been suspended since Mr. McCarthy’s ouster, including a fast-approaching Nov. 17 deadline for deciding on how to fund the federal government and a new request by the Biden administration for funding for the conflicts in Ukraine and the Middle East.

Electing the next speaker

The process of electing a new speaker is low-tech and transparent, as the world learned in January during Mr. McCarthy’s once-in-a-century floor fight to win the gavel. The entire House of Representatives gathers in the chamber, and lawmakers cast their votes in alphabetical order, by standing up and yelling out a name. Whoever earns a majority of those present and participating wins the race.

If the entire House is in attendance, that means a nominee needs at least 217 votes to be elected speaker. (There are currently 433 members of the House and two vacancies.) The math can change if there are absences, or if any lawmakers vote “present” rather than in support of a candidate.

If no one succeeds in meeting that threshold, the House simply continues to hold elections until someone does. Typically, a speaker has been elected after one floor vote. But if that proves impossible, the process can drag on indefinitely. Mr. McCarthy prevailed after five days and 15 votes.

Could Republicans form a coalition with Democrats?

The 212 Democrats in the House are expected to continue to vote as a united bloc for Representative Hakeem Jeffries of New York, the minority leader.

Mr. Jeffries has pitched the idea of forming a coalition government that he describes as an “enlightened arrangement.” But the idea is a long shot. And given that he has more votes than any Republican seeking the speakership, it is highly unlikely that Mr. Jeffries would agree to cede to a G.O.P. candidate without substantial concessions.

Mr. Jeffries said Democrats would team with Republicans to elect a speaker only if they agreed to change House rules to allow “governance by consensus” — in other words, allowing bills with bipartisan support to come to the floor. The Rules Committee, which determines what legislation gets a vote, is now structured so Republicans are in complete control of what bills the House considers. That means Democratic priorities are almost always blocked, and the hard right effectively has veto power on what is considered and what is not.

Mr. Jeffries said on Oct. 15 that “there are informal conversations that have been underway,” but he has since declined to offer any details about what a power-sharing agreement would look like.

Is the House still working without a speaker?

Legislative business in the House has been halted as Republicans struggle to unite behind a speaker. That includes work on legislation to fund the government and avoid a shutdown that will begin in mid-November if no action is taken. Also frozen is any consideration of an aid package to Israel, something that President Biden has said is an urgent priority after the terrorist group Hamas launched one of the broadest incursions into Israeli territory in 50 years.

Is there a way for the House to function without an elected speaker?

Representative Patrick T. McHenry of North Carolina is acting as the speaker pro tempore, a position created after the attacks of Sept. 11, 2001, to ensure continuity of government in case the speaker were killed or incapacitated. The position has never been tested, and so far, Mr. McHenry and House aides have interpreted the role very narrowly, simply as a place holder who presides over the election of a new speaker.

Some more centrist G.O.P. lawmakers had been working on a resolution that would explicitly grant Mr. McHenry the power to bring legislation to the floor, giving his inchoate role more clearly defined authority.

Doing so would require a vote.

Republicans debated taking an internal party vote on the matter last week, but a wide swath of lawmakers said they opposed such a measure, so it was ultimately shelved. Empowering Mr. McHenry, one of Mr. McCarthy’s closest allies, was regarded by many far-right members as akin to reinstalling Mr. McCarthy as speaker.

But the resolution could come up for a vote again, particularly if Republicans are unable to elect a speaker as the government shutdown deadline approaches. It is also not clear whether Democrats would support it, unless they secured commitments that their legislative priorities would be addressed.

Another option would be for Mr. McHenry to simply try to bring up a bill and, should a lawmaker challenge his power to do so, the issue would be put to a vote of the House. If an overwhelming majority was in favor of such a measure — for instance one providing aid to Israel or keeping government funding flowing to avert a shutdown — the House could act.

Luke Broadwater contributed reporting.