In an interview Monday morning (AEST) on US ABC's This Week with George Stephanopoulos, Mr Giuliani said Mr Trump reimbursed his personal lawyer Michael Cohen with his own money in a retainer agreement.
A secret $130,000 payment was made on the eve of the hotly contested 2016 U.S. presidential election to silence Daniels on claims of a decade-old affair.
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.
Giuliani, who has contradicted the president on when he learned of the payment, has said he's still learning the facts of the case.
WSJ: Investigators looking at Cohen's money sources during campaign
She pointed to the president's tweets as proof that the payment was not made with campaign dollars and was instead a means to "save the embarrassment for his family".
The Wednesday admission, which caught Mr. Trump's White House staff off guard, caused an uproar, and prompted Mr. Trump to attempt to clarify the nature of payments he made to Mr. Cohen.
Giuliani said on Friday that Cohen's payment to Daniels - one month before the November 2016 presidential election - did not violate campaign laws and would have been made even if Trump were not running.
"Now this was for another goal, to protect him, to protect his family", Giuliani said on ABC News.
Giuliani also told Stephanopoulos that he doesn't have knowledge of any payments to other women by former Trump lawyer Michael Cohen but that he couldn't rule it out.
Cummings' letter comes after Giuliani revealed to Fox News host Sean Hannity that Trump reimbursed Cohen for the $130,000 payment.
Avenatti also spoke to ABC, calling Giuliani's interview "an absolute, unmitigated disaster for Rudy Giuliani and the president, a train wreck".
Clinton, whom the FBI investigated for using a private email server to conduct official business as secretary of state, was interviewed without having been sworn in under oath, and no one prepared a transcript of the relatively brief discussion.
Daniels, whose legal name is Stephanie Clifford, is seeking to be released from the nondisclosure agreement. "I found this out many, many longtime afterwards long after I got into the White House", Conway said.
Speaking to reporters on Air Force One several weeks ago, Trump said he did not know about the payment or where the money came from. "I mean, what we witnessed by Rudy Giuliani may be one of the worst TV appearances by any attorney on behalf of a client in modern times".
For all the controversy Giuliani stirred up, some Trump supporters said it was wise to get the payment acknowledgement out in the open. "I have zero commitments, personally or professionally", Short said. Trump has denied the extramarital affair allegations made by Daniels and McDougal.
"I have no knowledge of that, but I would think if it was necessary, yes", Giuliani said.
Besides Daniels, former Playboy model Karen McDougal claims she was intimate with Trump "many dozens" of times from 2006 to 2007. "That's why these people who think they're chasing this down are involved in the same witch hunt that Judge Ellis was so concerned about with (ex-FBI director) Robert Mueller", he said.