He added that the president “didn’t know about the specifics of it, as far as I know, but he did know about the general arrangement that Michael would take care of things like this”.
Ms Daniels’ lawyer, Michael Avenatti, said that Americans “should be outraged” at Mr Giuliani’s comments.
“We predicted months ago that it would be proven that the American people had been lied to as to the $130k payment and what Mr Trump knew,” he wrote on Twitter.
After initially denying the payment, Mr Cohen eventually admitted he had paid the sum privately to Ms Daniels, real name Stephanie Clifford, in October 2016 out of his own funds in exchange for her silence.
He has denied that Mr Trump was a party to the transaction.
In a statement to the New York Times, Mr Cohen said neither the Trump campaign nor the Trump organisation knew anything about the payment, and he said he had not been repaid.