WASHINGTON—Within a day of the release of the Mueller report last month, President Trump sought to have former White House counsel Don McGahn declare he didn’t consider the president’s 2017 directive that he seek Robert Mueller’s dismissal to be obstruction of justice, but Mr. McGahn rebuffed the request, according to people familiar with the matter.

Mr. Trump has publicly denied asking Mr. McGahn to fire the Russia probe special counsel since the release of the report. Mr. Mueller’s report detailed that directive, and a subsequent...