Federal prosecutors are investigating Carl Icahn’s former role advising President Donald Trump and the activist investor’s attempts to change an environmental rule that he opposed.

Mr. Icahn, an early backer of Mr. Trump’s White House bid, was named special adviser to the president on overhauling federal regulations in December 2016. Mr. Icahn didn’t have an official government role, both he and the White House have said, but the appointment cemented his unofficial status as a sounding board for Mr. Trump.

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