The special counsel plans to finish by Sept. 1 its investigation into whether President Trump obstructed the Russia inquiry, according to the president’s lawyer Rudolph W. Giuliani, who said on Sunday that waiting any longer would risk improperly influencing voters in the midterm elections in November.

The office of the special counsel, Robert S. Mueller III, shared its timeline about two weeks ago amid negotiations over whether Mr. Trump will be questioned by investigators, Mr. Giuliani said, adding that Mr. Mueller’s office said the date was contingent on Mr. Trump agreeing to be interviewed. A spokesman for the special counsel’s office declined to comment.