A group wants the Justice Department and Office of Government Ethics to scrutinise "legal issues" related to the appointment of Jared Kushner, 36.
His lawyer says the post does not breach anti-nepotism laws.
In his new role as a senior White House adviser, Mr Kushner will initially focus on trade policy and the Middle East, according to officials from the transition team.
It should be mentioned that Kushner is married to Mr Trump's daughter Ivanka.