In the meantime, Trump's lawyers are preparing to oppose a potential subpoena from Mueller for a Trump sit-down by drafting a rebuttal that could set off a dramatic fight in federal courts.
"We would move to quash the subpoena," Giuliani said in an interview. "And we're pretty much finished with our memorandum opposing a subpoena."
Giuliani added that Trump's attorneys are ready to "argue it before the Supreme Court, if it ever got there."
In recent weeks, Giuliani said members of Trump's team have "had conversations" with Emmet Flood, a White House lawyer working on issues related to the federal investigation. He said Flood "would have a big role to play here and would assert presidential privilege" but declined to say more about those discussions.
White House officials have privately said Flood has cautioned Trump and others about the unpredictability of a subpoena fight that could be decided by the Supreme Court. Such a case would be unprecedented. Independent counsel Kenneth Starr served President Bill Clinton with a subpoena to compel him to appear before a grand jury, but it was withdrawn after Clinton agreed to testify voluntarily.
Mueller, who is heading the federal probe of Russian interference in the 2016 presidential campaign and of Trump's conduct, has sought a presidential interview for months. Negotiations between the president's lawyers and Mueller have failed to reach an agreement.
A letter from Trump's lawyers sent to Mueller on Aug. 8 significantly lessened the possibility of a voluntary presidential interview, according to two people familiar with the discussions. The multipage response represents what Trump's lawyers expect to be their last word on Mueller's request, according to the people, who spoke on the condition of anonymity because of the sensitive nature of the negotiations.
On a live radio program last week, Giuliani and Trump lawyer Jay Sekulow did not elaborate on their rejection of Mueller's interview terms, but both said the special counsel should finish his probe soon.
Giuliani acknowledged on Monday that the stalled talks probably frustrate federal investigators who are seeking to wrap up a report for the part of the probe that deals with Trump and possible obstruction of justice.
"There is always the chance that if they get fed up with our conditions, they just file their report as is," Giuliani said. "We're preparing a report to respond to that, should that be file," along with the response to a subpoena.
Mueller told Trump's lawyers in March that he could issue a subpoena for Trump to appear before a grand jury if Trump declined an interview, according to four people familiar with the encounter.
Giuliani said the Trump legal team, which briefed Trump last week, is mapping out a subpoena battle that could stretch on for months.
"Even if we responded in 10 days to a subpoena, it would have to be decided by a district court judge, and you could appeal it in a circuit court, and then you argue it before the Supreme Court, if it ever got there."
But for now, Giuliani and the president wait - and keep close watch of the federal trial of Trump's former campaign chairman Paul Manafort, who faces 18 charges of bank fraud and lying to the IRS that could send him to prison for the rest of his life.
"It's been a week, and we haven't heard anything," he said. "Plan A is we somehow work out the questioning. Plan B is we don't and they subpoena. Plan C is they file the report and don't issue a subpoena."
Giuliani said that the outcome of the Manafort trial could be a variable in the public perception of Mueller's investigation: "[Mueller's team] thinks they're empowered if they win and defeated if they lose."
When asked if Trump is mulling a pardon for Manafort, Giuliani said: "My position has been, 'I don't want to be part of the pardon process.' But my advice is, 'You shouldn't pardon anyone during the investigation or else you create a confusing situation.' He's bought into that."