Representative Tom Emmer of Minnesota, the No. 3 House Republican, emerged on Tuesday as his party’s latest leading contender for speaker as the G.O.P. ground through rounds of closed-door votes to break a deadlock that has left Congress paralyzed for three weeks.
Mr. Emmer won the first rounds of secret balloting, according to lawmakers who participated, and the field was winnowing as the lowest vote getters were forced out.
That left four Republicans, none of them with a national profile, vying for the post. Still in the race were Representatives Mike Johnson of Louisiana, a conservative lawyer who sits on the Judiciary Committee; Byron Donalds of Florida, a charismatic younger member of the ultraconservative House Freedom Caucus; and Kevin Hern of Oklahoma, the chairman of the conservative Republican Study Committee. The free-for-all reflected the deep divisions within the House G.O.P. The party began meeting Tuesday morning and was to remain cloistered behind closed doors for much of the day, grinding through multiple rounds of voting by secret ballot to try to coalesce around a candidate. Whoever ultimately wins a majority of Republicans will become the party’s next nominee for speaker and advance to the House floor.
Representatives Pete Sessions of Texas, the former chairman of the Rules Committee; Jack Bergman of Michigan, a retired Marine Corps lieutenant general; and Austin Scott of Georgia, who mounted a surprise challenge for speaker last week, were among those to be dropped on Tuesday after receiving the fewest votes.
Even before voting got underway, one candidate, Representative Gary Palmer of Alabama, withdrew. He said that the House needed a speaker three weeks ago and that “if withdrawing my name can help expedite that process even a little, then I will gladly step aside.”
If a nominee is chosen, a House floor vote could occur as soon as Tuesday afternoon, but there is no guarantee that the winner will have the 217 votes necessary to be elected, a threshold that has eluded the past two nominees.
“I don’t think anybody has that now,” Mr. Donalds said on Monday night. “I think we’re going to have to work to that.”
In a sign that continuing rifts could complicate Mr. Emmer’s path, lawmakers said a handful of Republicans were voting “present” or for other candidates who were not seeking the nomination. That suggested a potentially problematic group of holdouts who could deny him a majority.
The House has remained frozen since Oct. 3, when hard-right rebels forced a vote to oust Speaker Kevin McCarthy. Eight Republicans backed that move along with Democrats, who remained united behind their own leader, Representative Hakeem Jeffries of New York. In the weeks since, Republicans have tried and failed repeatedly to rally around a successor, even as wars rage overseas and a government shutdown approaches.
“While I wouldn’t have done the vacate, we’ve done it — here we are,” said Representative Chip Roy of Texas, referring to the motion to remove Mr. McCarthy. “The American people are seeing this, and we’re going through candidates, and we’re going to run and see who we want to get behind.”
Representative Mike Garcia of California, who represents a district won by President Biden, said he was supporting Mr. Emmer because he voted to keep the government open, unlike some of his competitors. Having a leader who “is not intentionally driving the government toward a shutdown is important to me,” Mr. Garcia said.
One lesser-known candidate, Representative Dan Meuser of Pennsylvania, dropped out on Monday evening as discussions began on the next nominee.
Mr. Meuser said his constituents were raging and wanted the House to get back to work.
“The American people — my constituents — are furious,” he said. “They are frustrated, they are angry. They’re not blaming just the eight, they’re not blaming Joe Biden. They’re blaming us, they’re blaming me.”
All of the candidates in the race except for two — Mr. Emmer and Mr. Scott — voted to object to certifying Mr. Biden’s 2020 victory in at least one state.
All but two of them — Mr. Hern and Mr. Johnson — voted in support of a stopgap spending bill put forward by Mr. McCarthy, the speaker at the time, to avert a shutdown. Mr. Donalds was absent for that vote.
Catie Edmondson, Robert Jimison and Kayla Guo contributed reporting.