But it's proving to be a bewildering display. "He (Cohen) made payments for the president or he's conducted business for the president".
Trump said Friday that he is still willing to talk directly with Special Counsel Robert Mueller but worries that he would be treated unfairly. He says: "I'm going to walk him into a prosecution for perjury like Martha Stewart?" Giuliani asked, referring to the lifestyle maven convicted in 2004 of lying to investigators and obstruction in an insider trading case.
Mr Giuliani couldn't guarantee that the president wouldn't end up asserting his constitutional right to refuse to answer any questions that might incriminate him.
"How could I ever be confident of that?" he said.
Conway said she has "no reason" not to believe Trump's comments denying the affair, and she denied that the White House has a problem with credibility. "The mob takes the Fifth Amendment".
Giuliani told ABC News a presidential pardon for Cohen is "not on the table" at the moment and has not been discussed amongst Trump's legal team.
A subpoena fight would likely find its way to the Supreme Court, which has never firmly decided whether presidents can be compelled to speak under oath.
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He revealed last week that Trump had reimbursed Cohen, a startling announcement that stood in direct contrast to the president's own public claims weeks earlier.
The FBI raided Cohen's home, office and hotel room last month in part to seek information that included payments made to women who alleged affairs with Trump more than a decade ago.
When asked on Sunday whether Trump knew about the payment to Daniels after the campaign, Giuliani demurred. "I can just say it's rumor", Giuliani said.
Asked by CNN anchor Jake Tapper on "State of the Union" whether she knows of any other payments similar to the one to Daniels, Conway said, "I do not".
Giuliani stated in an interview Wednesday evening on Fox Information that Trump reimbursed Cohen for the "so-called hush" cost that was made to Daniels previous to the 2016 presidential election. I hope we get a chance to tell him the risk that he's taking. "So he's saying he didn't know about it when the payment occurred. The reality is those are not facts that worry me as a lawyer".
"I have no knowledge of that, but I would think if it was necessary, yes", Giuliani replied.
Norman Eisen, ethics chief under Trump's predecessor, Democrat Barack Obama, said any such payments would have constituted campaign contributions if they were meant to influence the election, even if they were also made with the "partial motive of protecting the Trump family from embarrassment".