Investigators are expected to interview several other current and former White House officials in the coming days, including Mr. Trump’s former White House spokesman, Sean Spicer, and the White House communications director, Hope Hicks.

Mr. Mueller wants to question Mr. Spicer and Ms. Hicks about what occurred on an Air Force One flight in July, when Mr. Trump and his advisers deliberated about how to respond to questions from the news media about the June 2016 meeting between the campaign officials and the Russians.