Former national security adviser Mike Flynn is cooperating with the probe of Russian meddling in the 2016 election, pleading guilty Friday to lying about calls with Moscow’s ambassador a month before President Trump’s inauguration, contacts that prosecutors said were coordinated with senior members of the president’s transition team.

Jared Kushner, the president’s son-in-law, was one of those advisers who spoke with Mr. Flynn about at least one of the calls, said people familiar with the matter, bringing the long-simmering investigation closer to the president and his family.

In a 45-minute hearing in Washington federal court, Mr. Flynn admitted he misled FBI agents about a series of calls he had last December with the ambassador, Sergey Kislyak, about sanctions imposed on Russia by the Obama administration and about a United Nations resolution critical of Israel.

Prosecutors said Mr. Flynn consulted several times with top Trump transition officials on his contacts with Russia, suggesting special counsel Robert Mueller is closely scrutinizing other prominent Trump associates. But his ultimate targets, and the exact nature of any potential wrongdoing, remain unclear.

Mr. Flynn also acknowledged in court documents filed Friday that a “very senior member” of the transition team had directed his contacts over the U.N. issue. The people familiar with the matter said the “very senior member” was Mr. Kushner.

Abbe Lowell, a lawyer for Mr. Kushner, didn’t return multiple requests for comment.

Mr. Mueller is six months into a wide-ranging investigation into Russia’s efforts to meddle in the election and potential links between those efforts and the Trump campaign, which both Russia and Mr. Trump have denied. In court documents, Mr. Mueller’s office said Mr. Flynn’s lies to the FBI impeded that investigation.

When asked by U.S. District Judge Rudolph Contreras how he pleads to the charge against him, Mr. Flynn, flanked by his two lawyers, said softly, “Guilty, your honor.”

Given his cooperation, Mr. Flynn is expected to face at most six months in prison, depending on his “substantial assistance in the investigation or prosecution of another person who has committed an offense,” according to his plea agreement.

Prosecutors didn’t elaborate Friday on who else they may be looking at. Mr. Flynn pleaded guilty to one count of lying to the FBI, but a “statement of the offense” filed in connection with his plea detailed additional wrongdoing, including lying on a regulatory form about his private work directed by Turkish government officials. Mr. Mueller’s decision not to file more serious charges against Mr. Flynn led many legal experts to infer that Mr. Flynn is providing significant information.

“Mueller would not have accepted an agreement to plead to a single charge absent an arrangement of some sort, and presumably that is Flynn’s truthful testimony about other matters within Mueller’s jurisdiction,” said Paul Rosenzweig, a former federal prosecutor who was a deputy to special counsel Kenneth Starr during his investigation of President Bill Clinton.

In a statement Friday, Mr. Flynn said, “I recognize that the actions I acknowledged in court today were wrong, and, through my faith in God, I am working to set things right.” He added: “My guilty plea and agreement to cooperate with the Special Counsel’s Office reflect a decision I made in the best interests of my family and of our country. I accept full responsibility for my actions.”

Mr. Flynn, who was a high-profile campaign surrogate for Mr. Trump but who lasted in his White House post for less than a month, becomes the most senior Trump associate to face criminal charges in the special counsel investigation. At the 2016 Republican National Convention, Mr. Flynn led the crowd in chanting “Lock her up!” in reference to Democratic opponent Hillary Clinton and her use of a private email server while she was secretary of state. On Friday, protesters outside the courthouse chanted as Mr. Flynn left: “Lock him up!”

Documents filed in federal court Friday detail Mr. Flynn’s misstatements to the FBI during an interview last January and depict a group of incoming Trump officials concerned about the Obama administration’s new sanctions on Russia and trying to figure out how to react to them.

Mr. Flynn “falsely stated” to the FBI that he hadn’t asked the Russian ambassador to refrain from escalating a response to the sanctions, a senior prosecutor in Mr. Mueller’s office, Brandon Van Grack, said in court. In fact, Mr. Flynn had called a senior official of the Trump transition team on Dec. 29—the day the sanctions were announced—to discuss “what, if anything, to communicate to the Russian ambassador,” according to a statement filed in connection with Mr. Flynn’s plea.

A person familiar with the matter identified that senior transition official as K.T. McFarland, who was Mr. Flynn’s deputy at the time. A second person said that Ms. McFarland discussed with Mr. Flynn his conversation with the Russian ambassador.

The unnamed senior official was with “other senior members” of the transition team at Mar-a-Lago, Mr. Trump’s golf club in Florida, the document said. The official and Mr. Flynn discussed the fact that members of the transition team at Mar-a-Lago “did not want Russia to escalate the situation,” the document said.

Ms. McFarland met that day with Mr. Trump at Mar-a-Lago, along with Stephen Miller, Kellyanne Conway, Steve Bannon and Reince Priebus, a Trump spokesman said at the time. Mr. Miller and Ms. Conway are now serving in the White House. Mr. Bannon and Mr. Priebus served as chief strategist and chief of staff, respectively, but were ousted over the summer. Ms. McFarland is Mr. Trump’s nominee for ambassador to Singapore.

“Immediately after his phone call,” Mr. Flynn called Mr. Kislyak and asked that Russia only respond “in a reciprocal manner,” the document said.

The next day, Dec. 30, Russian President Vladimir Putin released a statement that Russia wouldn’t take retaliatory measures, and Mr. Kislyak called Mr. Flynn on Dec. 31 to tell him Russia had “chosen not to retaliate in response to Flynn’s request.”

After his phone call, according to the document, Mr. Flynn spoke to “senior members” of the transition about his conversations with Mr. Kislyak and about Russia’s decision “not to escalate the situation.”

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In addition to the sanctions issue, Mr. Flynn also lied about calls he made to “Russia and several other countries” about a U.N. resolution submitted by Egypt on Dec. 21, which criticized Israel’s construction of settlements in disputed territories.

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Mr. Flynn “falsely stated” that he only asked on the calls about the countries’ position on the votes, the document said. In fact, “a very senior member of the Presidential Transition Team”—Mr. Kushner, according to two people—“directed Flynn” to contact the officials to “influence those governments to delay the vote or defeat the resolution,” the document said.

The document also said Mr. Flynn made “materially false statements” in a regulatory filing about his separate work for Turkish interests, including stating that his company “did not know whether or the extent to which” the Turkish government was behind the project.

Mr. Flynn is the fourth—and most prominent—figure to be publicly charged in Mr. Mueller’s investigation to date. Former Trump campaign chairman Paul Manafort and his longtime associate who also worked with the campaign, Richard Gates, were indicted in October for prior work they did in Ukraine, unrelated to the Trump campaign. Another adviser, George Papadopoulos, has pleaded guilty to lying to the FBI about his contacts with Russian go-betweens.

Former FBI Director James Comey, whom Mr. Trump fired in May, has testified that Mr. Trump asked him to back off his investigation of Mr. Flynn, which the president denies.

Prosecutors’ references Friday to Mr. Flynn’s conversations with other Trump transition officials suggest Mr. Mueller is scrutinizing the actions of other high-profile figures in Mr. Trump’s orbit. Mr. Kushner said in a statement to Congress in July that, at Mr. Trump’s request, he served as the main point of contact for foreign countries during the transition.

Sen. Dianne Feinstein of California, the top Democrat on the Senate Judiciary Committee, said Friday’s plea shows that a Trump associate was “negotiating with the Russians against U.S. policy and interests before Donald Trump took office and after it was announced that Russia had interfered in our election.”

She added, “It’s critical that we determine whether Flynn spoke with the Russians on his own initiative and who knew and approved of his actions.”

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A statement from White House lawyer Ty Cobb played down the guilty plea, noting that Mr. Flynn was in the job less than a month and had “entered a guilty plea to a single count.”

Mr. Flynn’s false statements, Mr. Cobb added, “mirror” similar ones he made to Vice President Mike Pence about his Russian contacts earlier this year, which the White House has said resulted in his resignation. “The conclusion of this phase of the special counsel’s work demonstrates again that the special counsel is moving with all deliberate speed and clears the way for a prompt and reasonable conclusion,” Mr. Cobb said.

U.S. investigators began scrutinizing Mr. Flynn’s contacts with the Russian ambassador after the two spoke in a series of phone calls during the presidential transition in December 2016, according to people familiar with the investigation.

During those calls, Mr. Flynn discussed the potential easing of U.S. sanctions imposed on Russia after the Trump administration came to power in January, current and former officials said at the time, citing transcripts of intercepted communications between Mr. Flynn and Mr. Kislyak.

—Rebecca Ballhaus, Ben Kesling and Del Quentin Wilber contributed to this article.

Write to Aruna Viswanatha at Aruna.Viswanatha@wsj.com and Shane Harris at shane.harris@wsj.com