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.

“We want the concentration of this to be on Comey versus the president’s credibility, and I think we win that and people get that,” Mr. Giuliani said, adding that he also hoped that the Justice Department would open a criminal investigation into Mr. Comey for perjury and his role in the disclosures to The New York Times last year about his encounters with the president that prompted Mr. Mueller’s appointment.

Mr. Giuliani said Mr. Comey had damaged his credibility during his recent book tour and was an unreliable witness. Mr. Trump’s lawyers have long said that Mr. Comey’s accounts of their interactions, described in contemporaneous memos and congressional testimony, cannot be trusted. Mr. Comey declined to comment.

A month after Mr. Giuliani was hired to represent the president, his legal strategy is coming into focus. By assailing Mr. Comey, he is targeting perhaps the main witness in the obstruction investigation, a tack that Mr. Trump’s lawyers also tried over the past year. They told both the special counsel and the deputy attorney general, Rod J. Rosenstein, that Mr. Comey perjured himself and illegally leaked information, making the claims repeatedly in previously unreported letters described to The Times by several people who have seen them.

And by putting an end date on the obstruction inquiry, he is apparently seeking to publicly pressure Mr. Mueller to stick to that timeline and trying to assuage the president by predicting the inquiry will end soon, a strategy that some of his other lawyers tried, with mixed results.

The president, Mr. Giuliani said, wants any report summarizing the obstruction investigation to be made public.

His comments about the timeline echoed tweets by Mr. Trump hours earlier, when he complained that a prolonged inquiry would hurt Republicans in the midterms, citing a Times report over the weekend that Mr. Mueller was investigating offers of help to the Trump campaign from Saudi Arabia and the United Arab Emirates. It is illegal for foreign governments or individuals to be involved in an American election.

“Now that the Witch Hunt has given up on Russia and is looking at the rest of the World, they should easily be able to take it into the Mid-Term Elections where they can put some hurt on the Republican Party,” Mr. Trump wrote. “Don’t worry about Dems FISA Abuse, missing Emails or Fraudulent Dossier!”

Mr. Giuliani said the article showed that Mr. Mueller had found no coordination with Russia by Mr. Trump’s advisers and had undertaken a desperate bid to uncover conspiracies with other governments.

“It shows them at the end of the road — are we now going to check Africa and South America?” Mr. Giuliani said.

He said that he and Mr. Mueller’s office were still hammering out the terms of an interview with the president. He portrayed his client as a willing interview subject, saying that in the president’s view, no evidence exists that his associates coordinated with Russia’s election interference.

Mr. Giuliani said that an interview would be a distraction for the president and that the amount of preparation required meant that the president could not sit for questioning until after the scheduled summit meeting between Mr. Trump and the North Korean leader, Kim Jong-un, on June 12 in Singapore. Based on that schedule, Mr. Giuliani said, the president could be questioned around Independence Day.

The letters that Mr. Trump’s lawyers sent over the past year to Mr. Mueller and Mr. Rosenstein describe Mr. Comey as a witness lacking credibility and suggest that an obstruction case against the president had no basis. Two were written by Marc E. Kasowitz, Mr. Trump’s lead lawyer on the case at the time, within four days of each other in June 2017, and one on Jan. 29 by both John Dowd, who replaced Mr. Kasowitz and has since left the president’s legal team, and Jay Sekulow, who remains a part of it.

Mr. Trump’s lawyers said the special counsel had no basis to build an obstruction case based on Mr. Comey’s firing because the president was free as the head of the executive branch to dismiss any subordinates.

Mr. Comey, the lawyers said, was fired not to end the Russia investigation but because of how he handled the Clinton investigation. Mr. Rosenstein and Attorney General Jeff Sessions agreed with the president’s assessment, the lawyers said. Mr. Rosenstein, who is now overseeing Mr. Mueller’s investigation because Mr. Sessions recused himself, “actually helped to edit Mr. Comey’s termination letter and actively advised the president accordingly,” one letter said.

The lawyers did not say whether Mr. Trump had asked for an end to the Flynn investigation. But their letters cited statements by the White House that denied Mr. Comey’s account.

The lawyers also argued that Mr. Trump could not have impeded the investigation because there was no inquiry to obstruct. The letter said that the F.B.I. had concluded that Mr. Flynn had not committed a crime when he told agents in January 2017 that he did not discuss sanctions with the Russian ambassador during the transition, an assertion later found to be false.

The lawyers said that law enforcement officials had told the White House that the bureau did not believe Mr. Flynn had lied. “For all intents, purposes and appearances, the F.B.I. had accepted Flynn’s account; concluded that he was confused but truthful; decided not to investigate him further; and let him retain his clearance,” the letters said.

It is not clear what basis his lawyers have for those assertions. Mr. Flynn pleaded guilty in December to lying to investigators and is cooperating with the special counsel inquiry.

The letters also said that Mr. Comey had told Congress in a closed-door briefing in March 2017 that Mr. Flynn had not lied to the F.B.I. in the interview and was merely confused. Mr. Comey said last month on NBC’s “Meet the Press” that that assertion was not accurate.

The lawyers took issue with how Mr. Comey had described his interactions with Mr. Trump in the Oval Office. They noted that in congressional testimony after he was fired, Mr. Comey said that he did not push back on the president’s request, and they labeled his memos “self-serving notes.”

“The president had every right to express his view of the case,” the lawyers said, adding that “a president can also order the termination of an investigation by the Justice Department or F.B.I. at any time and for any reason.”

By firing Mr. Flynn, the lawyers said, Mr. Trump “actually facilitated the pursuit of justice.”

The lawyers also addressed another issue that has emerged about Mr. Trump’s behavior in office: his repeated criticisms — often on Twitter — of the F.B.I. and Justice Department. The lawyers said that Mr. Trump’s statements were simply him doing his job.

“It is also worth responding to the popular suggestion that the president’s public criticisms of the F.B.I. either constitutes obstruction or serves as evidence of obstruction,” the letter said. “Such criticism ignores the sacred responsibility of the president to hold his subordinates accountable — a function not unlike public congressional oversight hearings. After all, the F.B.I. is not above the law and we are now learning of the disappointing results of a lack of accountability in both the D.O.J. and F.B.I.”

Their arguments had limited success. The Justice Department officials responded by asking its inspector general to examine how the contents of Mr. Comey’s memos were disclosed.

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Giuliani Sees End of Inquiry On Obstruction
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