Giuliani told Fox News that Trump funneled the money for Daniels through Cohen's law firm over several months, by putting him on a retainer of $35,000 when "he was doing no work for the president."
The former New York City mayor said the payments did not come from Trump's 2016 campaign, and the reimbursement to Cohen was "perfectly legal." He added the president was "probably not aware" of the payment to Daniels at the time.
"He did know about the general arrangement that Michael would take care of things like this, like I take care of things like this for my clients. I don't burden them with every single thing that comes along," Giuliani said.
Trump, who had previously denied knowledge of the payment and said he was unaware of where the money came from, finally addressed the money on Twitter Thursday.
"[Cohen] received a monthly retainer, not from the campaign and having nothing to do with the campaign, from which he entered into, through reimbursement, a private contract between two parties, known as a non-disclosure agreement," Trump said in a tweet, adding such agreements are "very common among celebrities" and "people of wealth."
The president noted the agreement was in "full force and effect" and will be used against Daniels, whose legal name is Stephanie Clifford.
Trump said the agreement "was used to stop the false and extortionist accusations made by her about an affair despite already having signed a detailed letter admitting that there was no affair."
"This was a private agreement. Money from the campaign, or campaign contributions, played no roll in this transaction," Trump said.
The tweets marked rare statements from the president about Daniels.
Michael Avenatti, Daniels' attorney, thanked Fox News Thursday for having Giuliani on TV to make "damaging statements."
"Please do not stop helping our case week in and week out by having Mr. Trump and Mr. Giuliani appear and make damaging [statements]," Avenatti tweeted. "You are truly THE BEST; where can we send the gift basket?"
Daniels filed a lawsuit seeking to nullify the hush agreement arranged by Cohen and involving Trump in exchange for the $130,000 payment. The adult film star also filed a defamation lawsuit against Trump, over a composite sketch of a man who allegedly threatened her not to speak about the alleged affair.