Mr Kushner has volunteered to speak to the Senate Intelligence Committee, the White House said.
It is examining Russia’s alleged interference in last year’s election.
The US intelligence community believes alleged Russian hacking during the presidential election was done to help Mr Trump defeat Hillary Clinton.
Russia denied the allegations and President Trump has branded the story “fake news”.
There are two congressional investigations into the issue, plus an FBI one.
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Meanwhile, Democratic lawmakers have been calling for Republican Devin Nunes, chairman of the House Intelligence Committee, to withdraw himself from all further investigations into possible links between the Trump team and Russia.
House Democratic leader Nancy Pelosi said he had discredited himself by secretly visiting the White House to review intelligence documents, before announcing he had information that Mr Trump and his advisers may have been subjected to incidental surveillance during the election.
Mr Nunes’ spokesman said he had visited the White House as it was “a secure location where he could view the information provided by the source”.
However, Adam Schiff, the top Democrat on the intelligence committee, said the White House visit, which was kept from other committee members, showed Mr Nunes was too close to the administration to “lead a credible investigation”.
As tense as all these situations have been and will be going forward for the 36-year-old property developer, it’s Mr Kushner’s dealings with Russian officials that may put him in the sharpest spotlight.
The Russian story, given congressional inquiries and the ongoing FBI investigation, has the potential to be a ticking bomb within the Trump White House.
When Mr Kushner sits down with committee investigators he will, in effect, be speaking for the president – and his words could go a long way toward assuaging concerns or bringing the controversy one step closer to the president himself.
The meeting will probably be cordial for now. If Mr Kushner is at some point called to publicly testify before a congressional committee on the matter, things could get heated very quickly.
It will have sweeping powers to reform procedures, with technology and data a key area and the help of Apple CEO Tim Cook and Microsoft founder Bill Gates reportedly enlisted.
Mr Kushner, 36, told the newspaper the “government should be run like a great American company”.
He is a property investor and media executive who is married to Mr Trump’s daughter Ivanka.He already advises the president on foreign relations, and is said to have been influential in helping President Trump choose staff for his campaign and in government.