Any public declaration by Mr. Mueller about the president’s innocence would also be a clear sign that the special counsel’s investigation has not broadened significantly beyond last year’s presidential campaign to include a close scrutiny of any of Mr. Trump’s past business dealings with Russians.
Whether the strategy will work is another matter. The plan rests on the premise that Mr. Trump has done nothing wrong — something the president has repeatedly told his lawyers and said publicly — and some lawyers connected to the investigation say that Mr. Cobb has been too willing to take the president at his word. If the White House moves too hastily, they argue, materials could end up in Mr. Mueller’s hands that might damage the president and other administration officials.
Donald F. McGahn II, the White House counsel, previously expressed fears that the document production could set a bad precedent for future administrations. Mr. Cobb has told aides that the White House should move deliberately and carefully, but not drag its feet.
Others doubt that Mr. Mueller will publicly clear Mr. Trump anytime soon, even if the documents and interviews do not show that he committed a crime. Mr. Mueller is building cases against two of Mr. Trump’s former advisers, Paul J. Manafort and Michael T. Flynn. Should either man cooperate with investigators, it might change Mr. Mueller’s view of how Mr. Trump fits into the Russia investigation.
Nevertheless, the president’s advisers have concluded that this strategy represents their best chance to lift the cloud hanging over the administration.
“Good for them if they can pull it off,” said Barbara Van Gelder, a prominent Washington white-collar lawyer who served in the Justice Department with Mr. Mueller. She said he was highly unlikely to give the White House any assurances as long as the investigations into Mr. Manafort and Mr. Flynn were open.
“Mueller’s not going to make a statement,” she said, “because he’s not going to want to claw it back.”
Mr. Comey had similar concerns. While F.B.I. agents investigated whether Mr. Trump’s associates had any connection to Russian meddling in the 2016 presidential election, Mr. Comey privately told Mr. Trump and members of Congress that the president was not personally under investigation. At least twice, however, he refused requests by Mr. Trump to say so publicly. Mr. Comey later told Congress that he did not want to make a public declaration that he might have to amend after further investigation.
One administration official said it was not yet clear how the White House would make its request for Mr. Mueller to publicly exonerate Mr. Trump, and there have been delays in getting documents to the special counsel. The internal White House review of the documents is not yet complete, and Mr. Mueller does not plan to interview many key White House officials until his team has reviewed all the documents he requested.
Mr. Cobb and several White House lawyers have spent weeks reviewing documents related to numerous subjects, including Mr. Trump’s firing of Mr. Comey, and his role in July in drafting a misleading statement to The New York Times about Donald Trump Jr.’s meeting with a Russian lawyer in June 2016.
Mr. Mueller’s prosecutors have indicated that they plan to ask detailed questions about that statement, written aboard Air Force One, which withheld the purpose of the June 2016 meeting: to get damaging information about Hillary Clinton as part of the Russian government’s efforts to help Mr. Trump.
Some of Mr. Trump’s associates remain suspicious of the special counsel and his team of aggressive prosecutors.
“While it would be good to clear the air on this entire issue so that the president can focus on governing, it presupposes that Mueller is an honest broker, and that he would not take nothing and make it into something, which would be my concern,” said Roger J. Stone Jr., a longtime informal adviser.
Christopher Ruddy, the chief executive of Newsmax Media and a friend of Mr. Trump’s, said the White House should challenge Mr. Mueller if he investigates anything beyond whether Mr. Trump or his associates colluded with the Russian campaign to disrupt the election.
But White House officials say they do not want a repeat of some of the tactics used in the 1990s, when President Bill Clinton battled an independent counsel in court and in the media as the investigation dragged on for years. Since Mr. Cobb joined the White House, Mr. Trump’s team has significantly muted its criticism of Mr. Mueller.
One sign of the White House’s willingness to cooperate is that officials are strongly considering letting Mr. McGahn speak to investigators about his private conversations with Mr. Trump, according to two people with knowledge of the discussions. Many presidents insist that conversations with their White House counsel be kept secret under the principle of executive privilege, but Mr. McGahn is seen by some as a witness who could be helpful to Mr. Trump.
Mr. McGahn was involved in the discussions about firing Mr. Comey, and officials believe he would say that Mr. Trump was warned that firing Mr. Comey would only prolong the Russia investigation. Lawyers believe that would help make the case that Mr. Trump was not trying to obstruct justice when he fired the F.B.I. director.
Even as the White House pushes for a swift resolution of Mr. Mueller’s inquiry, administration officials are bracing for fallout from the investigations into Mr. Manafort and Mr. Flynn. Prosecutors have signaled that they intend to indict Mr. Manafort, the former chairman of the Trump campaign, who is under scrutiny for tax and foreign lobbying matters.
But lawyers in the case say they see no evidence yet that Mr. Manafort will face charges of conspiring with Russia to disrupt the election.
The White House also hopes for a favorable report from the Senate Intelligence Committee, which has been investigating Russian election interference and several related matters. The committee’s leaders said this week that they planned to release a public report about their findings.
That report could be finished before Mr. Mueller’s investigation. Although committee leaders said they would leave any criminal matters to the Justice Department, a determination by the committee that none of Mr. Trump’s associates assisted the Russian campaign would be a boon for the White House even if Mr. Mueller refuses to publicly clear Mr. Trump.
“They want them to write a report saying ‘no collusion,’” said Ms. Van Gelder, the defense lawyer. “And then they can let Mueller twist in the wind.”