WASHINGTON— Robert Mueller’s investigators are asking questions about Jared Kushner’s interactions with foreign leaders during the presidential transition, including his involvement in a dispute at the United Nations in December, in a sign of the expansive nature of the special counsel’s probe of Russia’s meddling in the election, according to people familiar with the matter.

The investigators have asked witnesses questions about the involvement of Mr. Kushner, President Donald Trump’s son-in-law and a senior White House adviser, in a controversy over a U.N. resolution passed Dec. 23 that condemned Israel’s construction of settlements in disputed territories, these people said.

Israeli officials had asked the incoming Trump administration to intervene to help block it. Mr. Trump posted a Facebook message the day before the U.N. vote—after he had been elected but before he had assumed office—saying the resolution put the Israelis in a difficult position and should be vetoed. Mr. Trump also held a phone conversation with Egyptian President Abdel Fattah Al Sisi, whose government had written a draft of the resolution. Egypt proceeded to call for the vote to be delayed, but the resolution passed the following day, with the Obama administration declining to block it.

Israeli officials said at the time that they began reaching out to senior leaders in the Trump transition team. Among those involved were Mr. Kushner and political strategist Stephen Bannon, according to people briefed on the exchanges.

The White House referred questions to Mr. Kushner’s attorney, Abbe Lowell, and to a White House lawyer.

The motivation for the Mueller team’s questions about the U.N. is unclear. Investigators typically ask a host of questions over the course of a probe, and inquiries don’t necessarily indicate suspicion. Mr. Kushner figures into several events that Mr. Mueller is investigating, including a June 2016 meeting with a Kremlin-linked lawyer at Trump Tower.

Mr. Kushner said in a July statement that the meeting was brief and a waste of time.

Investigators have also asked witnesses about Mr. Kushner’s role in arranging meetings or communication with foreign leaders during the transition, the people said. The special counsel’s mandate gives Mr. Mueller a broad directive to examine any matters arising from the Russia investigation.

The Mueller team’s questions come as investigators scrutinize Mr. Kushner for his initial omission of any foreign contacts from a government form required to obtain a security clearance, The Wall Street Journal previously reported. Mr. Kushner later updated the form at least three times to include what he has said were more than 100 contacts with more than 20 countries.

Mr. Kushner has said the initial omissions were an “administrative error.” His lawyer on Friday declined a request by the Senate Judiciary Committee to provide documents surrounding the submission of the form, and subsequent updates to it, saying they were confidential government records.

A 1799 law called the Logan Act also bars Americans from communicating with a foreign government to influence the government’s actions related to a dispute with the U.S., but no one has ever been successfully prosecuted under the law.

In a statement that Mr. Kushner gave to congressional committees in July, he wrote that at his father-in-law’s request, he served as the main point of contact for foreign countries at a time when it was clear Mr. Trump would be the Republican nominee.

The new details offer a window into the kinds of questions Mr. Mueller is asking as part of a six-month investigation that has been conducted largely behind closed doors. The special counsel and congressional investigators are probing whether Trump associates colluded with Russia in what U.S. intelligence deemed were Moscow’s efforts to interfere in the 2016 U.S. presidential election. Mr. Mueller has a broad mandate to examine any matters arising from that probe.

The president has denied any collusion by him or members of his campaign, and Moscow has denied election interference. Mr. Kushner, the president’s son-in-law and one of his most trusted advisers, has also denied collusion, and his lawyer has said he is cooperating with the investigations.

A spokesman for Mr. Mueller’s office declined to comment.

Another element of Mr. Mueller’s probe has focused on whether the president obstructed justice in the May firing of James Comey, the former director of the Federal Bureau of Investigation. Mr. Mueller’s prosecutors have asked witnesses detailed questions about Mr. Kushner’s views of Mr. Comey and whether Mr. Kushner was in favor of firing him or had staked out a position, said the people familiar with the matter.

Mr. Trump fired Mr. Comey, initially citing the former FBI director’s handling of an investigation into Hillary Clinton’s emails, and later saying the FBI’s Russia investigation was a factor in his decision.

Mr. Kushner pushed for Mr. Comey’s firing in discussions among the president and his top advisers, according to four people familiar with the matter.

One person said Mr. Kushner’s reasons for wanting Mr. Comey dismissed included that FBI agents unhappy with Mr. Comey would applaud the move and that Democrats would cheer because they were angry about Mr. Comey’s handling of the Clinton email investigation. Another person said top White House aides including Mr. Kushner viewed Mr. Comey as too unpredictable following his handling of the Clinton probe.

One aspect of Mr. Mueller’s probe concerns whether Mr. Comey’s firing was an attempt to obstruct justice, so it is possible he is asking about any Kushner role to get a clearer picture of events leading up to the dismissal.

Mr. Lowell said in an interview, “When the president made the decision to fire FBI Director Comey, Mr. Kushner supported it.” A White House attorney added that Mr. Kushner had “no meaningful role” in the decision: “There’s no apparent evidence of Jared’s involvement in any decision-making process having to do with Mr. Comey’s firing.”

Investigators have also asked questions about a meeting Mr. Kushner held during the presidential transition with Sergey Gorkov, chief executive of Russian state-owned Vnesheconombank, which was added to the U.S. sanctions list in 2014 as part of an effort to punish Moscow for its role in the Ukraine conflict.

In his written statement to Congress, Mr. Kushner said he met with Mr. Gorkov on Dec. 13 and didn’t discuss Russian sanctions imposed by the Obama administration, specific policies or any matters dealing with Mr. Kushner’s private business interests. Mr. Kushner said Mr. Gorkov largely discussed his bank and the Russian economy.

At the same time, congressional investigators are questioning the accuracy of Mr. Kushner’s statements in interviews with two committees earlier this year and whether he fully cooperated in turning over relevant records, according to public statements by and interviews with lawmakers.

In a letter to Mr. Kushner’s attorney last week, the top Republican and Democratic members of the Senate Judiciary Committee said Mr. Kushner hadn’t complied fully with an earlier request for documents.

Mr. Kushner’s lawyer, Mr. Lowell, disputed the assertion.

Top Democrats on the House and Senate intelligence committees want Mr. Kushner to return and answer more questions that have come up as new witnesses and documents emerge, according to public statements by and interviews with lawmakers.

In interviews with congressional panels in July, Mr. Kushner said he was unaware of any contact between the campaign and WikiLeaks, the online operation that last year published a trove of damaging Democratic emails that U.S. intelligence agencies concluded were stolen by Russian hackers, according to a person familiar with the matter.

But emails provided to the congressional committees by Donald Trump Jr. , the president’s son, revealed that the younger Mr. Trump forwarded to Mr. Kushner and other aides one exchange he had with the website, according to an email reviewed by the Journal. Mr. Kushner subsequently forwarded that message to another campaign aide, according to a letter to his lawyer from the Senate Judiciary Committee that was reviewed by the Journal.

Mr. Lowell said Mr. Kushner had “no contact” with WikiLeaks and that he didn’t respond to the email from the younger Mr. Trump.

Rep. Adam Schiff (D., Calif.), the top Democrat on the House Intelligence Committee, said in an interview that Mr. Kushner had been interviewed “prematurely,” when the committee was “not ready.”

“We didn’t have the advantage of documents that we would have wanted to ask [him] about,” he said.