Here are six prominent Republicans to watch in the House leadership upheaval after Rep. Eric Cantor's primary loss and resignation as majority leader. (Theresa Poulson/The Washington Post)

The race to be the next House majority leader began to crystallize Thursday as one leading conservative announced that he will not seek the position, while another top contender was asserting that he had all the votes he needed to win.

But allies of Majority Whip Kevin McCarthy (R-Calif.) were asserting momentum Thursday morning in the race to succeed House Majority Leader Eric Cantor (R-Va.), hoping to swiftly seal the deal. That confidence came as Rep. Pete Sessions (R-Tex.), chairman of the House Rules Committee, vowed again Thursday morning to mount a serious campaign for the job.

Sessions did so after Rep. Jeb Hensarling (R-Tex.), chairman of the House Financial Services Committee, said Thursday morning that he wouldn’t run for leader — a blow to the most conservative Republicans who saw him as their best chance to elevate a like-minded colleague into leadership.

“Jeb is out; I’m running for majority leader,” Sessions told The Washington Post. He spoke after a closed-door meeting with about 40 conservative Republicans, mostly from Southern states, whose votes he’ll need if he expects to mount a competitive challenge to McCarthy.

The results of the leadership race won’t be known until next Thursday, when House Republicans plan to meet behind closed doors to anoint new lieutenants for Speaker John A. Boehner (R-Ohio).

We asked some of the most conservative House Republicans what qualities they'd like to see in their next leaders. (Jackie Kucinich and Julie Percha/The Washington Post)

Boehner declined to weigh in on the future of his leadership team at his weekly on-camera news briefing Thursday morning.

“It’s important we resolve this issue in a fair amount of time so that we can do the work that we were elected to do,” he told reporters. He declined to say whether he would back McCarthy for majority leader.

“I’ve worked with all 434 other members of Congress before; I can work with whoever gets elected,” he said.

In a signal that McCarthy is quickly consolidating support, House Budget Committee Chairman Paul Ryan (R-Wis.), the GOP’s 2012 vice-presidential nominee, said he would support McCarthy.

“Jeb’s not in, and I was waiting for him” before weighing in, Ryan said, before adding: “Kevin McCarthy has my vote.”

Rep. Adam Kinzinger (R-Ill.), who is whipping up support for McCarthy, said he believes the Californian has secured the votes needed to win.

“He’s got this pretty wrapped up,” Kinzinger said. “I think he had this wrapped up pretty quick. . . . At the end of the day I think McCarthy has got this one over the top.”

But Rep. Tom Price (R-Ga.), a key conservative broker, declined Thursday to say whom he would endorse, reflecting some tensions in the GOP cloakroom as the campaign continues.

“It’s a secret ballot,” he repeated when pressed on his preference for leader. Price, who has long pushed for more conservatives in the leadership ranks and once ran for the conference’s fourth-ranking position, also declined to comment on McCarthy’s ideology.

As McCarthy seeks to lock up support, his loyalists have adopted a motto: Speed kills. They’re relying on McCarthy’s deep network of supporters, deputy whips and a paper-based scoring system that dates back to the House leadership races of the 1990s.

The 49-year-old California native enjoys cozy ties with many of the Republicans elected in the 2010 tea party wave. He is a strong fundraiser with a tireless work ethic and is seen as the candidate of the party’s establishment.

McCarthy launched his bid Wednesday after spending Tuesday night consoling Cantor and dozens of rattled colleagues by phone, according to aides. In the conversations, McCarthy told colleagues he wouldn’t make a decision about his political future until Cantor made his own.

Restless on Wednesday morning, McCarthy skipped breakfast and began to work in his office about 7 a.m. to start planning a campaign that he would launch a few hours later. At lunchtime, he left the U.S. Capitol to hold a fundraiser for Rep. Susan Brooks (R-Ind.) and returned to hold a series of calls and meetings once news leaked that Cantor would be stepping down.

McCarthy is relying on a tight-knit inner circle of colleagues to build support, a group including Reps. Richard Hudson (R-N.C.), Fred Upton (R-Mich.), Dave Camp (R-Mich.) and Harold Rogers (R-Ky.) — an influential array of names in the House GOP conference.

The team is using a numerical ranking system once used by former House majority leader Tom DeLay (R-Tex.) that McCarthy still believes is a smart way to track friends and foes. All 233 House Republicans are given a numerical ranking between one and five. A “1” means the colleague is a loyalist, while a “5” denotes a critic or someone who needs more convincing. Rankings for individual members are a closely guarded secret, aides said, but provide McCarthy with critical intelligence on who might need extra attention. Aides cautioned that the internal rankings would remain fluid up until votes are cast in the leadership race next Thursday.

On Thursday morning, McCarthy’s loyalists said they are taking the Sessions bid seriously, out of respect for the hard-charging Texan, but do not sense that he has a political organization or enough whips to help him.

Sessions began texting colleagues Tuesday night asking to speak with them about his bid for majority leader. On Thursday morning, he was defiant and brushed aside suggestions that McCarthy had wrapped it up.

“The process got fast, but it’s a process. I’m out there talking to delegations,” he told The Post. “I’m giving people a real view of how I’d lead. I’ve chaired the Rules Committee and the NRCC, and I’m looking to bring people in, sell our message and lead.”

The vote to replace Cantor next week is expected to unfold much like previous contests to fill top spots, according to senior GOP leadership aides. To outside observers, the contest will mirror the secrecy of a papal election and the elimination rules of a televised singing contest.

A candidate for majority leader must earn a majority of the votes of his 232 colleagues. If three or more candidates run for the same position and none earns a majority (hypothetically, a three-way race between McCarthy, Sessions and Hensarling), the lowest vote-getter drops out of the race, and members will re-vote.

If McCarthy ultimately prevails next Thursday, Boehner will immediately call a vote to fill the whip’s job, leaving no time for another prolonged campaigning period.

No matter who wins or loses, aides insist that the vote tally will never be known, except for the few members who count paper ballots and then destroy them.

Wesley Lowery and Paul Kane contributed to this report.