Avenatti in his interview later within the present asserted that Giuliani appeared to be suggesting that the president's lawyer had "a slush fund".
But Giuliani, whose statements since he recently joined Trump's legal team have created some confusion, also said that the president "wants to testify" and that "we may actually work things out with Bob Mueller", the special counsel.
He said that he and Jay Sekulow, another lawyer for Trump, were in agreement that the president should avoid speaking with Mueller.
Giuliani said he first made Trump aware of the payment shortly after joining the case but now says he doesn't know when Trump knew about it. Trump said Saturday that Giuliani needed to "get his facts straight" but insisted they weren't changing their story. He told the New York Times later that Trump reimbursed Cohen a total of $460,000 or $470,000 after the campaign was over, which he said included "incidental expenses".
"How can I ever be confident of that?" "I'd be living in some kind of unreal fantasy world where everyone tells the truth", Giuliani said.
'I would love to speak because we've done nothing wrong, ' he said.
Kellyanne Conway, a counselor to the president, said Sunday on CNN's State of the Union that Mr Trump meant that he didn't know about the payment at the time it was made, not at the time the question was asked. "I hope we get a chance to tell him the risk he's taking".
Giuliani, who said he had other TV appearances lined up to give away secret information, was then looped by Cohen. Baldwin's Trump asked Daniels.
Rudy Giuliani's statements are proving hard for the Trump administration. He has called Ms Daniels' allegations of an affair "false and extortionist".
Cohen, who acted as Trump's "fixer", is facing a criminal investigation in NY, and FBI agents raided his home and office several weeks ago seeking records about the Daniels nondisclosure agreement.
Michael Avenatti, Daniels' lawyer, said on This Week that he thinks it's "obvious.to the American people that this is a cover-up, that they are making it up as they go along". $130,000 cost when there have been different - I did clarify - and I don't know what they're but.
Conway also said she did not know of any other payments made to women during the campaign that were similar to the Daniels transaction, saying "they didn't cross my desk as campaign manager". Experts have said that even if it was not made with campaign money, the timing of it raises questions, as does the fact that it was never revealed in financial disclosure forms. "It may have involved the campaign, doesn't matter, and it would've been done under any other circumstances, if there was no campaign".