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...