The Washington Post, which has reported the move, describes the Office of Innovation as a “Swat team” of former private-sector executives.
It will have sweeping powers to reform government procedures, the report says.
Mr Kushner told the newspaper that the “government should be run like a great American company”.
“Our hope is that we can achieve successes and efficiency for our customers, who are the citizens,” he added.
Technology and data are expected to be a key area, with Apple CEO Tim Cooke and Microsoft founder Bill Gates said to be working with the White House.
In a statement to the Washington Post, President Trump said the new unit would fulfill key campaign promises.
“Government stagnation has hindered our ability to properly function, often creating widespread congestion and leading to cost overruns and delays,” Mr Trump said.
He said the new office would allow him to apply his “ahead of schedule, under budget mentality” to his government
Mr Kushner, 36, 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 chose staff for his campaign and in government.
Top White House strategist Steve Bannon is expected to have no formal role within the new group.
Mr Bannon’s previous calls for the “deconstruction of the administrative state” drew considerable attention.