After word leaked out about a clash at the White House where members of President Trump’s cabinet challenged the authority of Elon Musk to reshape their departments, one of the president’s top allies, Stephen K. Bannon, quickly piled on.

Mr. Bannon, who has characterized Mr. Musk as an interloper, a “parasitic illegal immigrant” and a “truly evil person,” suggested the world’s richest man was weighing Mr. Trump down.

“I don’t want to say an anchor or lodestone,” Mr. Bannon said on Friday of Mr. Musk on his show “War Room,” which is watched closely by a number of Trump allies, as well as the president himself. “It’s not that yet, but it’s trending — that is starting to affect everybody.”

The longstanding animus between Mr. Bannon and Mr. Musk encapsulates a key tension at the heart of Mr. Trump’s Make America Great Again movement. It pits those like Mr. Bannon, who want Mr. Trump to carry out a more fully populist agenda, against ultrawealthy interests, epitomized by Mr. Musk, who occupy key positions in the president’s orbit.

Mr. Trump has made clear he wants to keep both men and their allies within his movement, but Mr. Bannon’s vocal disdain for Mr. Musk has been noticed by the president. In mid-February, the president told Mr. Bannon that he wanted him to lay off the attacks on Mr. Musk and for the two men to sit down privately, according to two people familiar with the comments.

That meeting has not happened yet, and it is not clear when or if it will.

But Mr. Trump’s effort to mediate between the two men, which has not been previously reported, reflects the president’s awareness that Mr. Bannon has a powerful megaphone with key parts of the MAGA base.

Mr. Bannon has been preaching about populism since the Tea Party wave slowly started to remake the Republican Party in 2010. He and his acolytes see Mr. Musk as an opportunist with no ideological stake in the MAGA movement who only wants to advance his own interests.

But Mr. Bannon’s vision for the movement also has its fair share of critics for its alignment with right-wing nationalism, which he has labeled a “badge of honor.” “Let them call you racists,” Mr. Bannon told a far-right gathering in France in 2018. “Let them call you xenophobes. Let them call you nativists.”

Mr. Musk, who was a Trump critic for years before becoming one of his biggest benefactors, has seemingly given little thought to the MAGA movement and its future. Mr. Musk has been privately irritated by Mr. Bannon’s attacks at times, according to people in touch with him. But he has only rarely engaged with Mr. Bannon. “Bannon is a great talker, but not a great doer,” Mr. Musk posted on X, his social media platform, last month. “What did he get done this week? Nothing.”

In a statement, Karoline Leavitt, the White House press secretary, said: “We do not comment on private conversations that may or may not have occurred. President Trump is thrilled with DOGE’s historic work under Elon Musk, and he will continue to cut the waste, fraud, and abuse in our federal government on behalf of the American people.”

A spokeswoman for Mr. Musk did not respond to a request for comment.

Mr. Bannon’s attacks appear to be as much about attention as anything, although his allies say he is fighting for what he views as the soul of the MAGA movement. His supporters are quick to mention him as a potential candidate for president in 2028, noting that he finished in second place in the Conservative Political Action Conference’s 2028 Republican primary straw poll last month, though far behind Vice President JD Vance.

“Steve is looking down the pipe,” said Raheem Kassam, a close Bannon ally and the editor in chief of The National Pulse, a right-wing news site.

“He’s looking into the future, and he’s saying, ‘Oh no, there’s an atheistic, amoral, C.C.P.-aligned, unaccountable foreigner that’s going to be the head of the MAGA movement at some point,’” Mr. Kassam said, referring to the Chinese Communist Party, “and I think he’s right to express the concerns in the way he’s doing it.”

For now, though, the MAGA movement is personality-driven, devoted primarily to whatever Mr. Trump wants it to be regardless of the inherent contradictions. As he has taken control of the Republican Party over the last decade, Mr. Trump has led the quasi-populist movement by knitting together a wide range of factions and ideas, many of which conflict with each other.

Mr. Bannon served as chief strategist during Mr. Trump’s first term before an acrimonious split in 2017, but he was among the people most devoted to the lie that the 2020 election was stolen from Mr. Trump. Because of that, Mr. Bannon has credibility with some of the supporters Mr. Trump needs to be successful, and with Mr. Trump himself.

Mr. Bannon also has clear ideological disagreements with Mr. Musk, particularly over immigration. Mr. Bannon vigorously disagrees with Mr. Musk’s support for H-1B visas, which allow high-skilled individuals to work in America. Mr. Bannon has also warned that billionaires like Mr. Musk and other tech executives — many of whom supported Democrats before backing Mr. Trump — will abandon the MAGA movement.

“Bannon has been a dyed-in-the-wool conservative for his entire life, and he believes very strongly in these core values,” said Barry Bennett, a Republican strategist who worked on Mr. Trump’s first presidential campaign with Mr. Bannon. “He is always naturally suspicious of people who pop up and don’t have the pedigree that he has.”

Mr. Musk in some ways came out of left field as a die-hard supporter of Mr. Trump.

Throughout Mr. Trump’s first term, Mr. Musk had a lengthy list of critiques. He privately disparaged Mr. Trump’s tariff policies, his obsession with coal mines instead of gigafactories and his hard-line stances on immigration. More broadly, Mr. Musk railed against those who he believed had become tribal in their politics, according to people familiar with his comments.

In 2023, Mr. Musk signaled he would support the presidential campaign of Gov. Ron DeSantis of Florida before changing his mind as Mr. DeSantis’s campaign struggled out of the gate. Last month, Mr. Musk posted on social media that he loves Mr. Trump “as much as a straight man can love another man.”

Among the MAGA faithful, both men have deep fan bases. Mr. Bannon is one of the movement’s original content creators, dating back to his days overseeing Breitbart News and now through “The War Room.” Mr. Musk, meanwhile, has endeared himself to the movement through his transformation of X, formerly known as Twitter, into a hub of conservative activity.

“Musk is the volume button,” Mr. Bennett said. “You can say things now on Twitter and you can reach millions and millions of people where 10 years ago you would reach tens of thousands. He has made that available to us. There are a lot of people who are in the right of center movement who are very reliant on the medium to distribute their content. They are very grateful to him for allowing that to happen.”

The admiration for both men was on full display last month at the annual Conservative Political Action Conference, where Mr. Musk and Mr. Bannon each received a rapturous reception.

“They’re both very popular in the conservative movement,” said Mercedes Schlapp, who was Mr. Trump’s White House director of strategic communications during his first term and is married to Matt Schlapp, the chairman of the conference. “They were both warmly received at CPAC, where they put their differences aside and really addressed the president’s achievement.”

Mr. Musk made an unexpected appearance at the conference, brandishing a chain saw and sporting a black MAGA hat instead of the trademark red. (He described himself as “dark, gothic MAGA.”) In his CPAC interview, Mr. Musk acknowledged that he used to be “politically neutral” but leaned Democrat. He said he had switched to supporting Trump “when I realized I was a fool,” citing cancel culture and efforts to infringe on personal freedoms.

Mr. Bannon spoke directly after Mr. Musk, but he shied away from any direct confrontation. Instead, Mr. Bannon praised Mr. Musk as “Superman.” He also pointedly noted that history books “ain’t gonna remember me or Elon Musk or Tucker Carlson or Sean Hannity.”

“They’re gonna remember two things, Donald Trump and MAGA, OK?” he said.

In an interview with The New York Times last month, Mr. Bannon said there were fundamental differences between him and Mr. Musk.

“He’s still not a populist nationalist, he’s a globalist,” Mr. Bannon said of the tech billionaire. “He and I have a chasm that is probably insurmountable.”

As for Mr. Musk, his future intentions regarding politics are unclear. But one thing is not: Having been born in South Africa, he is ineligible to run for president — a fact Mr. Trump has noted publicly.