House Speaker Kevin McCarthy's allies are scrambling to head off a right-wing revolt to oust the California Republican from his job and turning to the constituency that now suddenly holds the cards: House Democrats.

One idea moderate Republicans are proposing to get Democrats on board with saving McCarthy, according to multiple lawmakers familiar with the discussions, is to revise the rules package that governs how the House operates -- including making it harder for any single member to force a vote to remove the speaker, a tool known as the "motion to vacate." They're also discussing making changes to the powerful House Rules Committee, which is responsible for how bills are brought to the floor, sources said.

The sales pitch to Democrats is less about protecting McCarthy and more about putting an end to the dysfunction that has gripped the House -- and allowing them to get back to the business of governing.

"If the (motion to vacate) becomes a regular occurrence, we will need to revisit the rules package in a bipartisan manner," Pennsylvania GOP Rep. Brian Fitzpatrick, co-chair of the bipartisan Problem Solvers Caucus, told CNN.

But moderate Republicans may have a tough time convincing the majority of Democrats, whose distrust for McCarthy runs deep and who were turned off by his last-minute decision to pass a 45-day funding bill that punts on Ukraine aid, narrowly averting a shutdown after refusing for weeks to look at bipartisan solutions. All but one Democrat ended up supporting the legislation.

"The Democrats are going to stick together and be a showcase of unity and teamwork compared to the chaos and division on the GOP side," Rep. Jamie Raskin, a Maryland Democrat, told CNN. "Fundamentally, pro-Trump Republicans have got to solve the problems that pro-Trump Republicans create. We can't be their babysitters."

Added another Democrat: "As Kevin faces this vote, he is still doing things that piss off Democrats."

Some Democrats have warned colleagues that entering into a rules-changing agreement could have unintended consequences.

"Do we want to purchase ownership in the outcome of Congress' actions in the next several months when we'd have responsibility but not control?" one member said.

GOP Rep. Matt Gaetz of Florida, who has warned for weeks he would move to boot McCarthy if the speaker worked with Democrats to fund the government, announced on CNN's "State of the Union" he will try to oust the California Republican as speaker this week. He can call for the vote himself, but it's a high bar to succeed, as he needs the majority of the House to back the effort. That means Democrats -- still reeling from the last 24 hours of Republican whiplash -- have another key decision to make: let McCarthy get booted from his job or cut a deal to save him.

Asked by CNN whether he would need Democratic votes to stay in power, McCarthy didn't answer directly and instead defended his handling of the stopgap bill. "There has to be an adult in the room," he said, saying anyone who wants to oust him should "bring it."

All eyes are on House Minority Leader Hakeem Jeffries, the new top Democrat in the House who is expected to lead the charge in making the consequential play call. Right now, there is no consensus among House Democrats on what to do or what to potentially extract from McCarthy in return for their vote, with Democratic leadership telling members in a "Dear Colleague" letter they'll discuss the issue as a caucus this week. Jeffries has been privately counseling members to keep their powder dry and avoid cutting any side deals, sources say.

But the majority of Democrats -- particularly in the Congressional Progressive Caucus -- are loath to bail out McCarthy, and don't believe they could trust him even if he offered them major concessions, according to multiple Democratic lawmakers and aides.

Democratic Rep. Rashida Tlaib of Michigan said in a statement to CNN that McCarthy needs to be held accountable for "pushing an extreme agenda and enabling extremists in his party" and therefore "by refusing to support a motion to vacate, we are endorsing this extremism, and that is something that the residents in my district will not stand for. The American people are tired of the fact that the GOP is incapable of governing."

Still, sources say it would take only a handful of moderate Democrats to vote "present" -- and there could be enough -- which would make it harder for McCarthy critics to succeed, or to vote to "table" the effort, essentially killing it and siding with McCarthy. Several have already indicated that's the route they would take.

In the end, though, the decision to throw McCarthy a lifeline might not come down to the perfect deal; it may simply come down to not wanting to throw the House into chaos and not knowing who might be waiting on the other side to ascend to the speakership.

What Democrats are weighing

For Gaetz to succeed, he will need the majority of the chamber on his side. So far, just one Republican -- Rep. Eli Crane of Arizona -- has publicly backed the motion to vacate. But sources say there are between one and two dozen other Republicans considering supporting the move.

Still, that means Gaetz -- who has been gauging Democratic support in recent days -- would need nearly every Democrat in his corner. And there are differing viewpoints among Democrats about how to proceed.

One Democratic lawmaker who supports ousting McCarthy told CNN, "I have no trust in, nor respect for, McCarthy. He isn't a trustworthy partner. He makes promises he doesn't keep and is never true to his word. Negotiations with him are meaningless."

The trust has eroded over time, beginning when McCarthy first cast blame on Trump for the January 6, 2021, insurrection at the US Capitol only to then meet with him in Florida weeks later. Democrats also have watched hard-fought bipartisan agreements, including the debt ceiling and the National Defense Authorization Act, be thrown aside by McCarthy in pursuit of winning over hardliners.

But House Democrats also acknowledge that they don't know another Republican who can get the 218 votes needed to clinch the speakership. And Democrats argue they could use this as an opportunity to extract key concessions, including a power-sharing agreement or policy promises -- such as a vote on Ukraine aid, ending the impeachment inquiry of President Joe Biden, or sticking to the bipartisan debt ceiling deal that could take showdowns over government funding off the negotiating table.

House Democratic leaders released a statement after they voted for the stopgap spending measure saying they "expect" McCarthy to put a bill on the floor to provide money to Ukraine; Biden suggested to reporters Sunday there was a "deal" with McCarthy.

But McCarthy's office would not say whether he gave any assurances to the White House or Democrats on Ukraine aid, saying in a statement that the speaker believes any Ukraine funding would need to be paired with border security provisions -- which he has been saying for weeks.

"Speaker McCarthy has been clear and consistent: we have an obligation to fix the crisis at the border and to ensure any request for further aid to Ukraine is matched with a sound strategy and accountability," a McCarthy spokesperson said in a statement to CNN. "The House will continue to discuss these challenges in the weeks to come."

And in a sign of how challenging that would be, GOP Rep. Marjorie Taylor Greene of Georgia warned McCarthy against such a move, writing on social media: "The majority of the majority does not support haphazardly funding war in Ukraine ... We elected our Speaker, not the Democrats, and our Speaker should not violate the Hastert Rule."

The chaos that could ensue if McCarthy is ousted would be partially beneficial to Democrats who want to highlight Republicans' inability to govern.

"We want their dysfunction to be on full display for the American people," one Democratic aide told CNN. "This is about proving that they cannot govern and winning back the House."

For now, House Democratic leadership is telling members not to make any decisions publicly and to wait for further guidance, according to a source familiar.

"My advice to my fellow Democrats is simple: Follow the leader," former House Speaker Nancy Pelosi said Sunday on CNN's "State of the Union" with Jake Tapper.

But while Jeffries has maintained a more cordial relationship with McCarthy than his predecessor, there have been recent signs of strain.

Twelve hours before the government was set to shut down, with still no clear plan from McCarthy, Jeffries and the speaker were slated to have a conversation about how the rest of the day would play out.

But their talk never happened, a source familiar with the plan told CNN. Instead, McCarthy surprised Democrats with his sudden decision to quickly move a stopgap bill, leaving them scrambling to read the legislation -- and forcing Jeffries to give a lengthy floor speech to buy themselves more time.