Giuliani's comments echo statements he made Friday to the New York Times in which he said the president's team would discuss an interview if the Mueller investigation "show [s] us the basis and that it's legitimate and that they have uncovered something".
Although Trump still "wants to testify", as Giuliani insisted, the likelihood of him speaking with Mueller looks increasingly slim.
Cohen's home, office and a hotel room in NY were raided by the FBI in April as part of an investigation into his business dealings.
"My wife, my daughter and my son have my first loyalty and always will", Cohen, a former longtime personal lawyer to Trump, told Stephanopoulos in the June 30 interview.
Rudy Giuliani, lawyer to President Donald Trump, said he'd consider allowing Trump to be questioned by special counsel Robert Mueller if Mueller can demonstrate that there's a factual basis for his Russian Federation investigation.
"We have no reason to believe he did anything wrong", the former New York City mayor said of Cohen.
The only legal risk for Mr. Trump, Giuliani told CBS News' Paula Reid, is lying to investigators, which Giuliani says could only be proven by a witness or document giving an alternate version of events.
"Zero. None", said Giuliani.
Giuliani made the television rounds Sunday morning, appearing on three different programs, as speculation grew louder last week that Cohen was inching closer toward facing criminal charges and possibly working out a deal with prosecutors. "One day I will answer", he said. I'd want to see if they're texting the same thing [as Strzok], and I'm willing to bet a dinner that they are'. "Is that the current condition", ABC News' "This Week" host George Stephanopoulos asked Giuliani, a member of Trump's legal team.
Kasowitz says Comey stonewalled the president's request to clear his name in order to "sustain an investigative cloud" over his head that would make it hard for Trump to fire him.
Cohen, who was known as Mr. Trump's fixer, is now under criminal investigation by the Justice Department.
"The reality is that there are biases that have to be explored surrounding Mueller", Giuliani added, specifically pointed to DOJ Inspector General Michael Horowitz's recently released report.
Petrillo did not immediately respond to a request for comment.
Also Sunday, Giuliani said he believes that Trump didn't know at the time that Michael Cohen, his former personal lawyer, paid porn star Stormy Daniels to keep silent about an alleged affair with Trump. "And then how could you expect us to just walk our client like a lamb going to the slaughter?"
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