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 alleged 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, before Mr. Trump took office, 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 then-President Barack Obama’s administration declining to block it.

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