A high-ranking California lawmaker, Kevin McCarthy, has emerged as the leading contender in the Republican contest to fill one of the top positions in the US Congress.
But some of his colleagues have complained he is not conservative enough and have urged others to jump into the race.
House Majority Whip McCarthy has been asking other lawmakers to support his bid to become House of Representatives majority leader to succeed Eric Cantor.
Cantor has been forced to step down after his shock primary election defeat to a little-known challenger from the "tea party" movement.
Representative Pete Sessions of Texas, who chairs the House Rules Committee, has also said he would run in the Thursday election for the number two post in the House.
McCarthy, the No 3 ranking House Republican who is in charge of lining up support for legislation, grabbed early momentum over Sessions by picking up some endorsements.
One was from Cantor, who will serve out the rest of his term.
Representative Lynn Westmoreland of Georgia, who attended a meeting of Republicans from southern states on Thursday, said he would back McCarthy because of his leadership experience, even though he believed a Republican from a southern state should hold one of the party's top House positions.
"I just think that Kevin will do a great job as a leader. He's kind of battle-tested being in that whip's position,"Westmoreland said.
Oklahoma Representative Tom Cole, also a McCarthy supporter, said the Californian was personally popular. "He tends to be a unifier not a divider," Cole said.
But both McCarthy and Sessions are seen as mainstream conservatives and allies of House Speaker John Boehner, leading to some grumbling from the party's right flank that leaders were moving too quickly to keep one of their candidates from running an effective campaign.
"We don't have the line-up of conservative, rule-of-law candidates in place. So we're asking for a delay in this vote, so that there's time for the conference to come to its senses and evaluate all the opportunities we have going forward," said Representative Steve King of Iowa, a tea party favourite.
King and Minnesota Representative Michele Bachmann, another tea party favourite, both said they would not back any candidate who favoured a path to citizenship - which they term "amnesty"- for immigrants who entered the US illegally.
In Cantor's Virginia district campaign, tea party challenger David Brat, a political novice and economics professor who portrayed Cantor as too soft on immigration reform, succeeded in toppling the House majority leader.
Idaho Representative Raul Labrador, prominent among conservatives willing to defy the Republican establishment, is believed to be getting support from other members to jump into the majority leader race.
Labrador abandoned bipartisan House talks on immigration last year and has said he does not think this year is the right time for the issue either.
The election represents a high-wire act for Boehner. He would like to see a new team installed that would help him move legislation and avoid fiscal crises, but one that also would make tea party supporters in Congress feel they had a voice.
If House conservatives remain angry over the fight to fill Cantor's post, they could challenge Boehner's bid to remain as speaker later this year. The Ohio Republican survived a challenge from the right after the 2012 congressional elections.