Elon Musk tried engaging with the Federal Trade Commission as the agency intensified an investigation into Twitter’s privacy and data practices, according to documents reviewed by The New York Times, in a sign of the billionaire’s hands-on approach to the inquiry.

Mr. Musk, who owns Twitter, asked to meet with Lina Khan, the F.T.C.’s chair, but was rebuffed, according to the documents. Mr. Musk made the attempt late last year, a person with knowledge of the matter said. In a Jan. 27 letter declining the meeting, Ms. Khan told a Twitter lawyer to focus on complying with investigators’ demands for information before she would consider meeting with Mr. Musk.

Mr. Musk spoke last month with the F.T.C.’s lone Republican commissioner, Christine Wilson, according to an email between the agency’s staff members and two people with knowledge of the matter. Ms. Wilson plans to leave the agency on Friday over what she has said are concerns about Ms. Khan’s leadership.

Mr. Musk’s outreach to the F.T.C. points to the gravity of the agency’s inquiry into Twitter. The investigation is focused on whether the social media company has adequate resources to protect its users’ privacy after Mr. Musk bought it last year and then laid off thousands of employees. The agency has separately sought to interview Mr. Musk for the investigation; the interview has not occurred, a person with knowledge of the matter said.

It is rare for chief executives at companies to try to meet with the F.T.C.’s chair and commissioners while an inquiry is underway. But such meetings sometimes occur when the executives hope to convince the agency’s top officials that they are committed to abiding by their promises to the F.T.C.

“If you thought you could simply brush it aside, and it was not a matter of great concern, you’d just ignore it,” said William Kovacic, a former F.T.C. chair. “If you think it’s important, that would be a reason to seek out a meeting.”

Mr. Musk, Ms. Wilson and a spokesman for the F.T.C. did not respond to requests for comment. A press request sent to Twitter triggered an autoreply email with a poop emoji.

The agency’s investigation is rooted in a 2011 settlement that Twitter struck with the F.T.C. over privacy concerns, which requires the company to maintain a user privacy program. Last May, Twitter paid a $150 million fine over alleged violations of that settlement; at the same time, the F.T.C. expanded the agreement’s terms to prohibit Twitter from using certain data to target ads and required it to offer specific security features.

Peiter Zatko, Twitter’s former security chief, said in a whistle-blower complaint made public in August that the company had failed to abide by the terms of the 2011 agreement, renewing inquiries from the F.T.C. The investigation expanded after Mr. Musk took over Twitter in October and conducted mass layoffs, which were compounded by the resignations of privacy and compliance executives.

After Mr. Musk requested to meet with Ms. Khan, she consulted with the enforcement division inside the F.T.C.’s consumer protection bureau, which has been leading the Twitter investigation, according to the email among agency staff members describing the situation. Acting on the enforcement team’s advice, Ms. Khan declined to meet with Mr. Musk at that time.

In Ms. Khan’s Jan. 27 letter to Twitter, she noted that the company was under investigation and had dragged its heels in providing documents to the F.T.C., delaying depositions with witnesses including Mr. Musk. She said she was “troubled by Twitter’s delays and the obstacles that these delays are creating for the F.T.C.’s investigation.”

“I recommend that Twitter appropriately prioritize its legal obligations to provide the requested information,” she wrote. “Once Twitter has fully complied with all F.T.C. requests, I will be happy to consider scheduling a meeting with Mr. Musk.”

The email among agency staff members discussing Mr. Musk’s outreach indicated that he spoke with Ms. Wilson, the Republican commissioner. They were joined on the call by James A. Kohm, the F.T.C. official who oversees investigations into whether companies are complying with their privacy settlements with the agency, according to the email.

Mr. Kohm did not respond to emails seeking comment.

Ms. Wilson has asked for copies of the agency’s letters to Twitter demanding information and documents related to its compliance with the privacy settlement, a staffer said in the email.

In February, Ms. Wilson said in a Wall Street Journal opinion piece that she would resign from the agency, citing Ms. Khan’s “disregard for the rule of law and due process and the way senior F.T.C. officials enable her.” The conversation with Mr. Musk occurred after that op-ed was published, two people with knowledge of the matter said.

Mr. Musk has forged relationships with Republicans, who have cheered his vision for a Twitter that is free of policies restricting what can be said on the platform. Earlier this year, he met with Representative Jim Jordan, Republican of Ohio and the leader of a subcommittee of the Republican-controlled House Judiciary Committee focused on “weaponization” of the federal government.

This month, that subcommittee released a report on the F.T.C.’s investigation into Twitter, accusing the agency of “orchestrating an aggressive campaign to harass Twitter.”

David Vladeck, a former director of the F.T.C.’s Bureau of Consumer Protection who served during the Obama administration, said that speaking with one commissioner who is not the chair was “not a smart strategy” for Mr. Musk, and that having a conversation with the agency’s lone Republican could look “more like politics.”

Mr. Musk has previously engaged directly with the members of the F.T.C., Mr. Kovacic said. After a group of aerospace companies announced a planned joint venture to challenge SpaceX, Mr. Musk’s rocket company, he met with F.T.C. commissioners including Mr. Kovacic in 2006 to raise competition concerns.

Mr. Vladeck said the F.T.C. could ultimately penalize Twitter for privacy violations again, potentially significantly more than its last $150 million fine.

As the F.T.C. investigation has intensified, Mr. Musk has moved ahead with changing some features of Twitter’s platform. He plans to remove verification check marks, the main way the social network confirms the identity of a user, from notable figures including celebrities and others on Saturday if they are not signed up for Twitter Blue, a subscription service that costs $8 a month.

Twitter also plans to charge organizations $1,000 a month to be verified, but will make exceptions for its top 500 advertisers and for the 10,000 most-followed organizations that have been previously verified, according to an internal document seen by The Times. All accounts that purchase check marks will be reviewed to make sure they are not impersonating someone, according to the document.

Ryan Mac contributed reporting. Kitty Bennett contributed research.