Mr. Musk was increasingly relying on a new adviser, a 34-year-old, Russian-born ex-professional gambler named

Igor Kurganov.

Mr. Kurganov spent some of the pandemic sleeping in Mr. Musk’s home, where they chatted late into the night about how the world’s richest person might use his fortune to help shape the planet through a giving strategy known as “effective altruism.”

Mr. Kurganov had no experience in finance or security but was suddenly a central figure in both areas for Mr. Musk. He had moved from London to Texas and replaced some of Mr. Musk’s protection detail with new hires of his own. Not long after, the Tesla CEO told Mr. Birchall that he was so taken by the younger man’s ideas that he wanted to leave him in charge of his charitable giving, dispersing funds from Mr. Musk’s vast private fortune, currently around $230 billion, as he saw fit.

“Elon,” Mr. Birchall told his boss, according to three people briefed afterward. “You can’t.”

The clash between the two men, and their dueling efforts to gain the upper hand with Mr. Musk, provides a peek into the often tumultuous private workings of Mr. Musk’s inner circle. As Mr. Musk’s breakneck, $44 billion bid for Twitter shows, the entrepreneur can be prone to quick decisions, ones that he can reverse just as quickly as he makes them. That deal is now in peril, with Mr. Musk trying to walk away and

Twitter

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suing him to complete it.

All the while he is egged on by a rotating crew of investors, underlings and ever-changing friends whose power and personal wealth is closely tied to their proximity to Mr. Musk, according to people who do business with him.

Two men in particular have pulled the Tesla CEO in opposing directions.

Mr. Birchall is an Eagle Scout and practicing Mormon who doesn’t smoke or drink and grew up traveling California as part of a song-and-dance troupe called “The Birchall Family Singers.” He declined to be interviewed through a representative of Mr. Musk’s foundation.

Mr. Kurganov is a high-roller with a reported more than $18 million in poker winnings, with long hair and beard and a peaceful demeanor. He has said in a podcast interview that he dropped out of college because he was smoking too much marijuana. He didn’t respond to requests for comment.

Last winter, Mr. Musk came to a compromise between the two. He allowed Mr. Kurganov to oversee $5.7 billion in Tesla shares that the CEO pledged late last year to eventually donate to charity. The rest of Mr. Musk’s overall fortune remained under Mr. Birchall’s purview.

That decision wasn’t, however, the end of the fracas.

This account is based on interviews with more than a dozen people close to Messrs. Musk, Birchall and Kurganov, including associates of the Musk Foundation and investors.

Ultimate fixer

For a man of immense resources, Mr. Musk is a minnow in the world of big-dollar philanthropy. Compared with other magnates such as

Microsoft Corp.

co-founder

Bill Gates

and

Amazon.com Inc.

founder

Jeff Bezos,

whose wealth, like Mr. Musk’s, is tied up in shares of their companies, Mr. Musk has said little concrete about his approach to giving. He didn’t respond to requests for comment.

Mr. Musk’s personal foundation gave away $23.6 million in fiscal 2020, the most recent year for which data is available, filings show. That represented roughly 0.02% of his net worth as of the end of that year, according to Forbes rankings.

Though Mr. Musk is known to favor his own counsel most of all, the man in charge of his foundation since 2016 has been Mr. Birchall, one of the longest standing in his inner circle.

The alliance reflects Mr. Birchall’s role as the ultimate fixer for Mr. Musk, taking care of matters large and small. In addition to handling billions in assets, Mr. Birchall in 2018 under a pseudonym registered the website www.justballs.com, for example, after Mr. Musk mused online that he might want to own the domain name someday, legal documents show.

Mr. Birchall entered the technology world through finance. After his brief singing stint with his siblings—he is one of 11 children—he attended Brigham Young University.

He then spent a year at Merrill Lynch, where he was discharged for what the firm described in a filing as “sending correspondence to a client without management approval.”

Mr. Birchall eventually moved into private wealth and crossed paths with Mr. Musk about a decade ago while working as a client adviser in

Morgan Stanley’s

southern California offices. There, he impressed Mr. Musk by helping arrange hundreds of millions of dollars of loans from the investment bank at a time when the Tesla co-founder was strapped for cash, a person close to Mr. Musk says. When Mr. Musk started a private firm to manage his own money roughly six years ago, he tapped Mr. Birchall to run it.

Mr. Birchall’s work for Mr. Musk has at times led him into strange public controversies and lawsuits. After a British spelunker criticized Mr. Musk’s offer to help rescue a youth soccer team trapped in a flooded Thailand cave in 2018, Mr. Birchall created a fake email address under the name “James Brickhouse” to hire a private investigator to investigate the man, according to court documents.

Mr. Musk subsequently called the man “pedo guy” in a tweet, though he later apologized and said it was a joking taunt. The spelunker unsuccessfully sued Mr. Musk for defamation in a Los Angeles federal court.

In addition to his role helping manage Mr. Musk’s fortune, Mr. Birchall is also a director of his tunneling startup, Boring Co., and chief executive officer of Mr. Musk’s Neuralink, which aims to make a brain implant that could be used by quadriplegic patients to control a computer or other devices.

When Mr. Musk announced in 2020 he would move from California to Texas, Mr. Birchall moved his family there, too. He has since become the closest thing Mr. Musk has to a public face in the Texas capital, handling business and attending events on Mr. Musk’s behalf.

“No one else you reach out to who is close to Elon is going to return your calls,” says Tyson Tuttle, former chief executive of Austin semiconductor manufacturer Silicon Valley Labs.

‘Make big moves’

Mr. Kurganov first became friendly with Mr. Musk socially.

Born in Russia but raised in Germany from age 4 by parents who were engineers by trade, Mr. Kurganov grew up in a working class, immigrant household, according to an interview he gave to the Paul Phua Poker podcast.

In his 20s, he gained a level of fame as a professional poker player. Beginning around a decade ago, he began playing high-stakes tournaments in Las Vegas and racking up wins, fellow poker players say. He earned particular renown in 2012 when he won a €1,080,000 purse in Monte Carlo.

He was known as an aggressive player, and had the respect of his fellow pros, the players say. “Igor had a willingness to play a lot of hands and make big moves,” says one of them, Dan Smith.

Though he was spending much of his time in Las Vegas, his personality didn’t match the city’s loud stereotypes. One person who knows him says he was often drawn into long, philosophical conversations on topics such as the rise of artificial intelligence in poker. Mr. Kurganov was concerned that a computer program could be built that could beat even the most talented human player.

Ging Masinda, a Las Vegas marketer and acquaintance of Mr. Kurganov, says the flashiest thing about him is that he sometimes wears eyeliner. “There’s a kid in him,” she says.

In a 2015 interview on the Poker Life Podcast, he said he was developing an accounting app for poker players. He also co-founded the organization Raising for Effective Giving, designed to help poker players, as well as fantasy-sports players and finance professionals, find the right charities to which to donate their winnings, according to archived webpages.

REG was focused on effective altruism, an approach to giving that suggests that inherently subjective qualities such as relative charitable need can be quantified. It’s still a niche approach, because many charitable experts say it can encourage an impersonal, utilitarian approach to complicated moral issues.

Mr. Kurganov’s social life, the fellow poker players say, revolved around his longtime partner, fellow professional poker player Liv Boeree. Ms. Boeree has long been friends with the recording artist Grimes, who was first romantically linked to Mr. Musk in 2018. Around the same time, Mr. Musk made Ms. Boeree one of the handful of people he follows on Twitter.

“She was elated” about the follow, according to Joe Stapleton, a poker commentator who spoke to her about it.

The two couples began spending time together. Messrs. Musk and Kurganov bonded over a shared love of Burning Man, the free-spirited desert festival that both regularly attend. Ms. Boeree, meanwhile, stayed close with Grimes; the two women posted photos together on social media. Ms. Boeree didn’t respond to requests for comment, and Grimes, whose legal name is Claire Boucher, couldn’t be reached.

The introduction to Mr. Musk came as Mr. Kurganov was soon to retire from professional poker with millions in personal winnings. Yet his poker accounting app had never gotten off the ground, and REG remained small-fry in terms of the amount of money it directed.

REG said on its website that it directed $3.1 million in charitable giving in 2019, the most recent year mentioned, which the organization said “reflects all donations that have been significantly influenced by us.” The amount couldn’t be independently verified. REG didn’t respond to requests for comment.

When the pandemic hit, Messrs. Kurganov and Musk became closer. Mr. Kurganov laid out a vision in which he might be able to help Mr. Musk combine his interest in science and his vast fortune, for the greater good.

Mr. Musk, known to be open to new ideas, was intrigued by the prospect of an approach to giving that no other major donor had yet tried, according to people familiar with the matter. He gave Mr. Kurganov a job vetting grant requests from his foundation.

When Mr. Musk moved to Texas, Mr. Kurganov and Ms. Boeree soon followed.

‘Donation decisions’

By last August, two years after retiring from poker, Mr. Kurganov had an official email address at the Musk Foundation and was heavily involved in the charity. That month, he corresponded with Chris Carberry, CEO of the nonprofit Explore Mars Inc. The group was looking to raise $600,000 to spirit an Afghan girls robotics team out of the war-torn country.

Mr. Kurganov was skeptical about the cost. “Regarding the price breakdown, what I’m trying to find out is what the expenses will be specifically—I imagine there are additional costs to simply the flight costs, but am still surprised that the total cost scales linearly with the number of people,” he wrote in an email reviewed by The Wall Street Journal.

Mr. Carberry, who didn’t respond to requests for comment, was disappointed but not dissuaded. In another email, also reviewed by the Journal, he wrote to an associate “We have worked effectively with Igor in the recent past. Igor makes the day-to-day donation decisions.”

To Mr. Birchall, that was an untenable situation. He was legally head of the foundation, and as he saw it, Mr. Kurganov was a newcomer who suddenly had immense influence on what to do with Mr. Musk’s money, he told people.

Mr. Birchall also learned that a Federal Bureau of Investigation agent had begun making preliminary inquiries into Mr. Kurganov as part of his job to watch for foreign interference in U.S. companies, people familiar with the matter say. The FBI agent was concerned that a newcomer had so quickly been welcomed into Mr. Musk’s inner circle, the people say.

Mr. Kurganov hasn’t been accused of wrongdoing. An FBI spokesperson didn’t respond to a request for comment.

Mr. Birchall, alarmed that Mr. Musk could be drawn into a federal investigation, this spring again pressed his internal case against Mr. Kurganov, people familiar with the matter say. It was inappropriate, in Mr. Birchall’s view, for Mr. Kurganov to have a central role in the $5.7 billion in shares that Mr. Musk had promised to donate.

During this period, Mr. Birchall was a constant presence at Mr. Musk’s side as his boss pursued Twitter. He helped line up billions of dollars in financing and spoke to shareholders and investors on Mr. Musk’s behalf about the future of the social-networking service.

By spring, the Tesla CEO’s bid for Twitter required him to put up tens of billions of dollars in personal funding and was consuming much of his attention. Meanwhile, Tesla’s stock was dropping as the broader market pulled back, pinching Mr. Musk’s net worth.

Mr. Birchall in May asked Mr. Musk to remove Mr. Kurganov from his post at the foundation.

Mr. Musk agreed to let Mr. Kurganov go, the representative of the foundation says.

None of Mr. Musk’s money was ultimately spent on projects related to effective altruism, the representative says, and Mr. Kurganov’s Musk Foundation email was switched off roughly six weeks ago.

—Elisa Cho and Rebecca Elliott contributed to this article.

Write to Rob Copeland at rob.copeland@wsj.com