Mr. Giuliani appeared to be in part trying to shift responsibility onto the special counsel for the lengthy negotiations over an interview — and was most likely prolonging them himself.

“If they can come to us and show us the basis and that it’s legitimate and that they have uncovered something, we can go from there and assess their objectivity,” Mr. Giuliani said in an interview. He urged the special counsel to wrap up his inquiry and write an investigative report. He said Mr. Trump’s lawyers planned to write their own summary of the case.

A spokesman for the special counsel’s office declined to comment.

The president’s lawyers want Mr. Mueller to explain how the Justice Department gave him the authority to investigate possible obstruction of justice by the president in what began as a counterintelligence investigation into Russia’s election meddling. The order appointing Mr. Mueller authorized him to investigate possible links between Moscow’s interference and Trump associates, as well as any matters that arose from the inquiry.

The lawyers also want evidence that the special counsel exhausted every other investigative measure before asking the president to answer questions, and that he is the only person who could provide them with the information they are seeking.

The gambit by Mr. Giuliani was the latest maneuver in an all-out assault by the president and his legal team in recent months to alter public opinion about the inquiry. They have come to believe that, if the Democrats win control of the House in November, the chamber will vote on whether to begin the impeachment process no matter the outcome of Mr. Mueller’s investigation. So they want to sway Americans — and by extension, lawmakers.