WASHINGTON — White House officials once debated a scorched-earth strategy of publicly criticizing and undercutting Robert S. Mueller III, the special counsel investigating Russian efforts to disrupt last year’s election. Now, President Trump’s lawyers are pursuing a different course: cooperating with the special counsel in the hope that Mr. Mueller will declare in the coming months that Mr. Trump is not a target of the Russia inquiry.

Mr. Trump has long sought such a public declaration. He fired his F.B.I. director, James B. Comey, in May after Mr. Comey refused to say openly that Mr. Trump was not under investigation.

The president’s legal team is working swiftly to respond to requests from Mr. Mueller for emails, documents and memos, and will make White House officials available for interviews. Once Mr. Mueller has combed through the evidence, Mr. Trump’s lawyers plan to ask him to affirm that Mr. Trump is not under investigation, either for colluding with Russian operatives or for trying to obstruct justice.

More than a half dozen White House officials, witnesses and outside lawyers connected to the Russia inquiry have described the approach, which is as much a public relations strategy as a legal one. The president’s legal team aims to argue that the White House has nothing to hide, hoping to shift the burden to Mr. Mueller to move quickly to wrap up an investigation that has consumed the Trump administration’s first year.