“We certainly won’t do [an interview] after Sept. 1, because we’re not going to be the ones to interfere with the election,” Mr. Giuliani said. “Let him [Mr. Mueller] get all the bad publicity and the attacks for that.”

Based on a prior meeting with Mr. Mueller, Mr. Giuliani said he had believed prosecutors wanted to wrap up the inquiry by September. “Now they’re not really rushing us,” he said.

Mr. Mueller has made some moves that suggest the inquiry itself could stretch beyond the midterm elections and certainly past the September timeline Mr. Giuliani laid out.