The FBI is looking into potential Russian meddling in the 2016 election and links with Mr Trump's campaign. The president denies any collusion.
Mr Kushner's lawyer said his client would co-operate with any inquiry.
President Trump has described the situation as "the single greatest witch hunt of a politician in American history".
US intelligence agencies believe Moscow tried to tip the election in favour of the Republican, who beat his Democratic rival Hillary Clinton.
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Separately, the Washington Post reported that the investigators were focusing on meetings he held last year with the Russian ambassador to the US, Sergei Kislyak, and a banker from Moscow.
Robert Mueller, a former FBI boss, last week was named by the justice department as special counsel to oversee the Russia inquiry.
Congress is also looking into Russia's alleged meddling in the 2016 election and any Trump campaign ties.
Mr Kushner has already agreed to discuss his Russian contacts with the Senate Intelligence Committee.
"Mr Kushner previously volunteered to share with Congress what he knows about these meetings," Mr Kusnher's lawyer Jamie Gorelick told the BBC.
"He will do the same if he is contacted in connection with any other inquiry," the lawyer added.
Calls for a special investigation have mounted since President Trump fired the most recent FBI director, James Comey, earlier this month.
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Mr Flynn was forced out in February after he misled the vice-president about his conversations with Russia's ambassador before Mr Trump took office in January.
Russia has repeatedly denied any involvement in the US presidential election.
[BBC Sources]
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