The battle to win the House speakership could hinge on a contingent of more moderate Republicans uneasy about the conservative politics of their two leading candidates and angry at their hardline faction for ousting Kevin McCarthy in this week's unprecedented floor vote.

Behind the scenes, House Majority Leader Steve Scalise of Louisiana and House Judiciary Chairman Jim Jordan of Ohio have been making a direct pitch to more centrist members, both insisting they will make their reelection battles a priority and ensure more stability atop the badly divided conference, according to sources familiar with the conversations.

Jordan and Scalise have made overtures to a bloc of New York Republicans who are among the most vulnerable in the House and would be considered a key prize for whichever candidate wins them over. Scalise plans to meet virtually on Friday with freshman GOP lawmakers and a separate group of more mainstream Republicans, after Jordan made his case to the latter group during a call on Thursday.

"They have to lay out for us a real understanding that members like me not only deserve a seat at the table, but our voices have to be represented and heard," New York GOP Rep. Marc Molinaro, who has not endorsed a candidate yet, said of Scalise and Jordan. "And so, can they rise to that challenge?"

Jordan is also targeting another key constituency that is seen as up for grabs: moderate members who were loyal to McCarthy. Over the years, the former speaker has had a tepid relationship with Scalise, his top deputy, as the two were long seen as potential rivals. McCarthy, though, is staying neutral in the race right now.

Rep. Carlos Gimenez of Florida, a McCarthy ally who considers himself center right, said Jordan "certainly can win me over."

"He was one of Kevin McCarthy's most vocal supporters," Gimenez told CNN. "And you know what? That carries a lot of weight."

Jordan may soon have one key endorsement. Former President Donald Trump has privately told those around him that he wants to endorse the Ohio Republican for speaker and believes he is the best person for the job, a source close to the former president told CNN on Thursday. While his backing could boost Jordan's support on the right, it won't do much to sway moderates.

But many members are keeping their powder dry, worried that neither candidate would be able to win the support of nearly every Republican on the floor. That means the House could see a prolonged race on the floor -- much like in January -- and some members are holding out hope for a dark-horse candidate to emerge, potentially the interim speaker, Patrick McHenry of North Carolina, who so far hasn't taken steps for a possible run.

Shutdown, motion to vacate become top tier issues

At the center of the demands from many of the more moderate Republicans: winning assurances that the government doesn't shut down in mid-November while changing the rules so one member can't do what Rep. Matt Gaetz of Florida successfully did by calling for a vote and successfully seeking the ouster of a sitting speaker.

"I want to see conference rules change," said Rep. Max Miller, a freshman from Ohio. "I want to see the motion to vacate go back up to a majority of the majority at 112."

Asked if he could commit to backing a candidate who doesn't commit to changing the powerful rule, Miller said: "It's going to be very tough."

Rep. Garret Graves of Louisiana told CNN he wants to discuss the rule changes before holding speaker elections, telling reporters on Wednesday, "I think before we have a single discussion about the speaker, we've got to discuss the functionality of the position."

During his call with the business-minded Main Street Caucus on Thursday, Jordan also said any changes to the motion to vacate would need to be a conference decision, a source on the call said, but sounded open to changing it. The Main Street Caucus is pushing to make that mechanism harder to use. Under current rules -- which McCarthy agreed to when he was trying to win the job in January -- any one member can use it to oust a speaker. Some hardliners are insisting the rule stay in place.

Moderates weigh their options

Several Republicans believe it will be hard for either candidate to get the needed votes, particularly in such a short time frame. Some hardliners are skeptical of Scalise, while some moderates, especially in Biden-won districts, are worried about Jordan's flame-throwing brand of politics. Jordan is one of the faces of the Biden impeachment inquiry, which some of the moderates have been openly skeptical of.

One moderate Republican said they are "not anxious to support Jordan" and "Jordan's challenge is that he built the monster that's part of the problem."

One Republican from a swing district raised concerns about Jordan's opposition to more funding for Ukraine and his vote against a stopgap funding bill to keep the government open, "which means he wanted a shutdown." This member said: "I'm concerned about his stance on Ukraine. Republicans can't pull the rug out from, while it's defending itself from a Russian invasion."

Another swing-district Republican, however, said they're on the fence. They argued Scalise has been "very good to the swing-district Republicans" and "has a strong sensitivity for what majority-makers need politically," arguing there is a "lot of good will there."

But the member also said they "want to get to (yes)" on Jordan, though they said he would be a riskier choice. The lawmaker also said they might actually want to keep the motion to vacate in place so they could have a tool to keep a potential Speaker Jordan in check.

"I may want this low threshold on the motion to vacate if a speaker doesn't bring good governance," the member told CNN.

There's also discussion among some of the swing-district Republicans about voting together or making demands as a bloc, mimicking the playbook of the hardliners who ousted McCarthy.

"We know that if we stick together and assert our priorities, we're likely to be accommodated," the member said.

How fundraising could be a key factor

During the Main Street Caucus meeting on Thursday, Jordan promised to protect moderates and vowed not to put them in tough positions, according to a source on the call.

Rep. Nick LaLota, one of the vulnerable New York Republicans, spoke up on the call and noted that the Freedom Caucus tends to listen to Jordan, who helped found the group. LaLota has not endorsed any candidate yet.

In private conversations with some New York Republicans, sources say, Jordan has emphasized he cares about the issues that matter to them, noting he held his field hearing on crime in New York City and is also a fierce advocate for border security. Jordan has also pitched his conservative background as an advantage to moderates, arguing that his ties to the GOP hardliners make him the best positioned candidate to bring the divided conference together.

But Scalise could have a major fundraising advantage with the New Yorkers: He has campaigned in the districts of every single vulnerable Republican -- a point he has raised in his private conversations with members.

In the last election cycle, Scalise -- who has the advantage of having been in the upper ranks of leadership for years -- has raised over $53 million, while Jordan -- who only recently became Judiciary chair -- has raised more than $14 million, according to Federal Election Commission data. Meanwhile, Scalise allies point out, the political arm of the House Freedom Caucus, which Jordan has raised millions for, has supported primary challenges to 10 Republican incumbents over the past few cycles.

A source familiar told CNN that Scalise's team has repeatedly asked Jordan to send multiple campaign surrogate mailers -- which some of Jordan's allies say shows that Scalise recognizes his Ohio colleague has fundraising appeal.

In his pitch to members, sources say Scalise has argued that he has the experience, staff operation and institutional knowledge to govern the GOP conference. And as the former head of the Republican Study Committee, the largest conservative group in Congress, Scalise could have more credibility with the right than McCarthy did.

Whoever becomes speaker will also need to address a looming government funding deadline in mid-November to avert a shutdown. On a stopgap spending bill, Jordan said he would not work with Democrats but thought they could do a better job on messaging it. McCarthy's opponents have cited the California Republican's reliance on Democratic votes to avert a shutdown last weekend as one of the reasons for ousting him.

The House GOP conference is scheduled to have an in-person meeting Monday evening, according to a source familiar, and then a candidate forum Tuesday in hopes of holding elections for the next speaker in a full House floor vote as early as Wednesday, though that timing could slip if no candidate unifies the Republican lawmakers.

To win the speakership next week, the candidate first needs to win a majority of the 221-member House GOP Conference in a secret-ballot election, followed by a majority of the full House, which currently stands at 433 members with two vacancies. The winning candidate needs to secure 217 votes if all are present and voting, meaning the Republican can only afford to lose four votes.

"Getting to (217) is a very steep climb and it ought to be," Molinaro said.