WASHINGTON — The special counsel hopes to finish by Sept. 1 the 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 November’s midterm elections.

Mr. Giuliani said that 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, adding that Mr. Mueller’s office said that the date was contingent on Mr. Trump’s sitting for an interview. A spokesman for the special counsel’s office declined to comment.

Wrapping up the obstruction case would not signal the end of Mr. Mueller’s work. That is one piece of his broader inquiry, a counterintelligence investigation into Russia’s campaign to interfere in the 2016 presidential election and whether Trump associates coordinated with it. Counterintelligence investigations are used to gather information quietly about the activities of foreign powers and their agents — sometimes for years — and can result in criminal charges.

Mr. Giuliani sought to frame the outcome of the obstruction investigation as pitting the credibility of one man against another: Mr. Trump vs. James B. Comey, the former F.B.I. director, whom the president asked to end the investigation into his first national security adviser, Michael T. Flynn. The president’s request is one of the main episodes Mr. Mueller is examining to determine whether Mr. Trump had criminal intent to obstruct the Russia investigation.