House Republicans took their most concrete and public step this week toward pursuing the impeachment of Homeland Security Secretary Alejandro Mayorkas. But behind the scenes, supporters of the effort are still working to convince key holdouts to get on board with the potentially divisive plan.

House Homeland Security Chairman Mark Green on Wednesday outlined a five-phase investigation into Mayorkas and his handling of the border, accompanied by a preliminary report and a hearing focused on framing Mayorkas' "dereliction of duty," which starts a roughly three-month countdown for House Republican leadership and their allies to get the House GOP conference in alignment.

That lobbying effort starts with Republicans on the House Judiciary Committee, who would be responsible for formalizing Green's investigative findings into official impeachment proceedings. So far, however, not every GOP member on the panel is sold on the idea.

"We don't have the votes," House Judiciary Committee member Rep. Matt Gaetz of Florida said of his GOP colleagues on the panel.

GOP Rep. Andy Biggs of Arizona, another committee member, told CNN he is concentrating his lobbying efforts on a small clutch of key Republican lawmakers, though he declined to name names.

"There's probably two or three people that I'm trying to work on," he said, adding, "and if those two or three come along, then I think we're ready."

Among the Republican skeptics on the Judiciary Committee are Reps. Tom McClintock of California, Ken Buck of Colorado and Tom Tiffany of Wisconsin.

"Maladministration, even of this magnitude, is not grounds for impeachment," McClintock, who leads the subcommittee Rep. Jim Jordan has tapped to hold more hearings on Mayorkas, told CNN. "Show me treason, bribery or other high crimes or misdemeanors. And I'm all ears. But at the moment I have not heard it or seen it."

Buck, a former federal prosecutor and member of the hardline Freedom Caucus, said, "You've got to meet the high crimes or misdemeanors." And Tiffany echoed a similar sentiment, saying: "If this would happen, I want to see all the evidence and see if it rises to that level that is appropriate for impeachment."

House Judiciary Republicans, who are holding a hearing with Mayorkas at the end of July, meet weekly and the topic of impeachment often comes up, multiple sources told CNN.

The hope from outspoken supporters of a Mayorkas impeachment is that the Homeland Security Committee hearings over the next few months will help build momentum in the GOP conference.

"I think these Homeland hearings will definitely help anybody that isn't there yet get on board," said GOP Rep. Marjorie Taylor Greene of Georgia, who also has launched her own articles of impeachment against Mayorkas, and was silenced from speaking in a hearing in April for calling Mayorkas a liar.

Achieving unity among Republicans on the Judiciary Committee is only part of the battle, though. House Speaker Kevin McCarthy would still have to get the rest of his conference to support rare impeachment proceedings against a Cabinet secretary, which would be dead on arrival in the Senate. And while some moderates in key swing districts have started to warm up to the idea, others are still skeptical.

GOP Rep. Clay Higgins of Louisiana, who has filed his own articles of impeachment against Mayorkas and serves on the Homeland Security Committee, said broadly of the conference, "There are votes. Whether or not there is enough, that's another question."

That means the next few months could put already frayed relationships inside the fractious GOP conference to the test, with McCarthy under increasing pressure to cater to his right flank after the debt ceiling deal without upsetting his so-called majority makers. Even just passing a border security package -- a messaging bill designed to fulfill a key campaign promise -- proved exceedingly difficult. That's why GOP leadership and key committee chairmen have considered carefully how they discuss the topic of impeachment publicly.

Still, some GOP lawmakers are confident they will eventually get there.

"Ultimately, I think it's the speaker's call," said Rep. Mike Johnson of Louisiana, a member of GOP leadership who also sits on the Judiciary panel. "But I think it's inevitable."

'It will be a long process'

In the various impeachment articles against Mayorkas that have already been filed, Republicans have accused Mayorkas of undermining the operational control of the southern border, encouraging illegal immigration and lying to Congress that the border was secure -- all charges that the administration has dismissed.

Legal experts have also thrown cold water on the Republican basis for impeaching Mayorkas, arguing that it appears to be over political disagreements and doesn't rise to the level of high crimes and misdemeanors.

House Republicans have anchored their criticisms in President Joe Biden's decision to scrap Trump-era border policies, but the Biden administration has since put in place restrictive measures that, in effect, limit who's eligible for asylum at the border. Homeland Security officials have pointed toward a dramatic drop in border crossings since the end of a Covid-era border restriction, known as Title 42. Since May 11, when the authority expired, border crossings have dropped more than 70%.

"Secretary Mayorkas is proud to advance the noble mission of the Department, support its extraordinary workforce, and serve the American people," a Homeland Security spokesperson said in a statement. "Instead of pointing fingers and pursuing a baseless impeachment, Congress should work with the Department and pass comprehensive legislation to fix our broken immigration system, which has not been updated in decades."

The top Democrat on the Homeland Security panel, Rep. Bennie Thompson of Mississippi, criticized his Republican colleagues' impeachment efforts at Wednesday's hearing by saying, "It's about trying to make good on GOP backroom deals to elect a speaker, raise the debt ceiling and stave off a mutiny in the Republican rank."

During negotiations on the debt ceiling, Greene said she would swallow the deal, but told reporters she wanted "dessert" to go along with it -- and specifically named impeachment as her idea of a sweetener.

McCarthy, however, has stopped short of endorsing the idea publicly and repeatedly said they won't use impeachment for political purposes, though he has said Mayorkas ought to resign.

"I think I told you long before I even got this job, he hasn't been doing his job," McCarthy recently told CNN. "I mean, who in America believes the border is secure? Even when he says that, you can look in his face, that he's not being honest."

While Jordan and GOP leadership have a private understanding that the impeachment of Mayorkas is inevitable, as CNN has previously reported, McCarthy has told members he wants more time for committees to hold hearings on the subject before they move on impeachment articles.

While both the Homeland and Judiciary committees have already held multiple hearings on the border, the panels dedicated a lot of their time over the last few months trying to draft a border security package that could win support across the conference, which ultimately passed the House in May.

House Majority Whip Tom Emmer, a Minnesota Republican, told CNN that committees have now been instructed to move forward with their investigations towards impeachment and "We will see where it goes."

"I've thought that impeachment is certainly warranted based on (Mayorkas') terrible performance, but that's a question for the entire conference," Jordan told CNN. "It will be a long process. It comes to dominate everything. And maybe that's where we need to go. But it's a decision I think we have to make as a full Republican conference."

Green said it was too soon to jump to the question of impeachment, telling CNN, "We haven't even gotten to that word, that process. Our goal here is we are starting with what we have here and we're going to get more information about the failures of this secretary. And when we're done, we'll make a recommendation to the Judiciary Committee one way or the other and we'll see what they have to say."

While a number of Republicans attempt to corral their colleagues towards supporting impeachment, Democrats both on and off Capitol Hill are coordinating their counteroffensive effort.

The left-leaning outside group, Congressional Integrity Project, is gearing up to run an impeachment specific war room and coordinate closely with House Democrats to counter the GOP narrative on Mayorkas, a source familiar told CNN.