Reports say investigators believe Jared Kushner has relevant information, but he is not necessarily suspected of a crime.
The FBI is looking into potential Russian meddling in the last year’s election and links with Donald Trump’s campaign. The president denies any collusion.
Jared Kushner’s lawyer said his client would co-operate with any inquiry.
Donald Trump has described the Russia investigations as “the single greatest witch hunt of a politician in American history”.
US intelligence agencies believe Russia tried to tip the election in favor of Donald Trump, who beat his Democratic rival Hillary Clinton.
US officials, who were not named, told NBC News that the interest in Jared Kushner did not mean the investigators suspected him of a crime or intended to charge him.
Separately, the Washington Post reported that the investigators were focusing on meetings Jared Kushner held last year with the Russian ambassador to the US, Sergei Kislyak, and a banker from Moscow, Sergei Gorkov.
Sergei Gorkov is the head of Vnesheconombank, which has been subject to sanctions imposed by the Obama administration in response to Russia’s annexation of Crimea and its support for separatists in eastern Ukraine.
Vnesheconombank is under the control of Russian PM Dmitri Medvedev and other members of the government, and has been used to fund major projects such as the 2014 Winter Olympics in the southern Russian resort of Sochi.
Jared Kushner, 36, has said he did not discuss sanctions with Sergei Gorkov.
Last week, former FBI Director Robert Mueller was named by the justice department as special counsel to oversee the Russia inquiry.
Congress is also looking into Moscow’s alleged meddling in the 2016 election and any Trump campaign ties.
Jared Kushner has already agreed to discuss his Russian contacts with the Senate Intelligence Committee.
Calls for a special investigation have mounted since President Trump fired the most recent FBI director, James Comey, earlier this month.
The White House has been engulfed in crisis over allegations that President Trump asked James Comey to drop an inquiry into links between his ex-national security adviser Michael Flynn and Russia.
Michael Flynn was forced out in February after he misled the vice-president about his conversations with Russia’s ambassador before Donald Trump took office in January.
Russia has repeatedly denied any involvement in the US presidential election.
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