Regardless of how much of the report is eventually disclosed, Democrats are also pushing for information and material Mr. Mueller gathered but did not put in the report. Such files could include documents and records of interviews with witnesses, as well as any internal memos the Mueller team wrote analyzing negative information about people whom it ultimately decided against indicting.

The Justice Department traditionally opposes turning over such files — especially if they contain information relevant to investigations that remain open. Mr. Mueller has handed off many cases his office developed to regular Justice Department prosecutors for continued pursuit, and there remains a permanently open counterintelligence investigation into Russia.

“We’re definitely on a collision course for a major fight at both ends of Pennsylvania Avenue over both the Mueller report, testimony related to it and the underlying case files,” said Andrew Wright, who specialized in executive privilege matters as a White House lawyer in the Obama administration and was a congressional oversight staff member.