In recent weeks, Mr. Mueller’s office sent a document to the White House that detailed 13 different areas that investigators want more information about. Since then, administration lawyers have been scouring White House emails and asking officials whether they have other documents or notes that may pertain to Mueller’s requests. Mueller has also requested documents about the circumstances of the firing of Michael T. Flynn, who was Mr. Trump’s first national security adviser. Additionally, the special counsel has asked for documents about how the White House responded to questions from The Times about a June 2016 meeting at Trump Tower. That meeting was set up by Donald Trump Jr., the president’s eldest son, to get derogatory information from Russians about Hillary Clinton.