The affair took place in 2006, while Trump was married to his current wife and shortly after the birth of his youngest son, Baron.
He also called raids of Cohen's home, office and hotel room earlier this year "inconceivable" and "almost unheard of".
The "break in" referenced by Trump was actually the raiding by Federal Bureau of Investigation agents in April of the Manhattan office and residences of Trump's then-personal attorney, Michael Cohen, who is being investigated for potential bank and election-law crimes.
Giuliani said the discussion of payment did not mean McDougal's claim of an affair was true and characterized it as an attempt to resolve false allegations that were "personally damaging" to Trump. "In the big scheme of things, it's powerful exculpatory evidence". In New York state, it is legal to record a conversation if one party consents.
He cited two recent cases that show "despite the burgeoning use of surreptitious recordings by members of the public, there is little tolerance for lawyers who - albeit legally - surreptitiously record" other parties without justification.
The White House has said Trump denies having sex with Daniels.
Trump touts success of Putin summit despite continuing criticism
After his walk-back, Trump said his administration would "move aggressively" to repel efforts to interfere in American elections. The bill lays out specific acts by foreign governments that constitute election interference. "It's really clear", he said.
The development was first reported Friday by The New York Times.
But his attorney, Lanny Davis, dismissed any notion that Cohen could be put in legal jeopardy due to the recording. Trump's campaign had said it knew nothing about any payment to ex-centerfold Karen McDougal.
The search warrant needed for the raid had been issued thanks to a referral from the head of "Russiagate" probe, Robert S. Mueller, and has already been decried by Cohen's lawyer as "inappropriate and unnecessary". "Any attempt at spin cannot change what is on the tape". It was not immediately clear where Trump and Cohen were located at the time of the call.
Cohen had made a practice of recording conversations, unbeknownst to those he was speaking with.
"Obviously, there is an ongoing investigation, and we are sensitive to that", Davis said.
In her lawsuit against AMI, McDougal said she was happy when AMI bought her story then did not publish it, because she was not anxious for publicity.
In August 2016, AMI, the parent company of the National Enquirer, paid McDougal $150,000 for the right to her story but never published an article based on her account.
The matter is a potential problem for Mr Trump because undisclosed payments to bury embarrassing stories about a political candidate can be treated as a violation of United States campaign finance laws. But the agreement meant that McDougal signed a non-disclosure agreement that prevented her from revealing the affair.