Investigators believe Mr Kushner, 36, has significant information relevant to their inquiry, US officials told NBC News.
The FBI is looking into potential links between Mr Trump's campaign team and Russia during the 2016 elections.
Mr Trump denies any collusion between his campaign and Russia.
The US officials, who were not named, said that the interest in Mr 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 a series of meetings he held with the Russian ambassador to the US, Sergei Kislyak, and a banker from Moscow.
Congress is also looking into potential links between Mr Trump's campaign team and Russia.