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.