Mr Kushner, a senior White House aide, has not commented.
He is said to be under scrutiny by the FBI as part of its inquiry into Russian interference favouring Mr Trump’s win.
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Reports in the US say investigators believe he has relevant information, but he is not necessarily suspected of a crime.
What did Kushner do?
The most recent reports – which cited unnamed US officials as sources – said Mr Kushner had spoken with Moscow’s Ambassador to the US, Sergei Kislyak, about setting up a back channel using Russian diplomatic facilities in America.
The meeting was held in early December at Trump Tower in New York – Mr Trump’s power base.
According to both reports, Mr Trump’s first national security adviser Michael Flynn was present at the meeting.
The secret channel was supposed to be used to discuss Syria and other policy issues during the transition period between Mr Trump’s election in November and his inauguration in January 2017.
The Washington Post says the proposal surprised Mr Kislyak as it meant Americans using Russian facilities at their diplomatic missions in the US.
The New York Times said the line was never established.