The two discuss polling numbers and the impact of a supportive African-American pastor.
"I did it on my own", Cohen said, according to the Journal.
Before the election, Trump's campaign denied any knowledge of payment to McDougal, but the taped conversation could undermine those denials.
In the tape, Cohen suggested to Trump they buy the rights to former Playboy model Karen McDougal's story, which had been bought by the publisher of the National Enquirer a month earlier. The company paid McDougal for her story about her rumored affair with the president but never published it, effectively gagging McDougal.
In a tape released by CNN, US President Donald Trump can be heard discussing with his personal lawyer how to buy the rights to the story of a Playboy model who claimed she had an affair with him. Weisselberg has detailed information about the Trump Organization's operations, business deals and finances.
Weisselberg also served as treasurer of the president's troubled philanthropy, The Donald J. Trump Foundation, which the New York State Attorney General has sued for allegations involving "extensive and persistent violations of state and federal law".
"It clearly indicates that the president did not want to pay with cash".
If Trump made the alleged payments with his own funds, and made the payments strictly for personal, not campaign reasons, he could be legally in the clear. On one level, it's hardly a smoking gun since the payment that Trump and Cohen are discussing never actually happened.
The conversation took place after The Journal in January revealed that Cohen had arranged the payment to Daniels.
The tape becomes hard to understand at that point.
Trump can then be heard saying the word "cash" but because of the muddled audio, it is unclear what precedes it.
"Pay with cash", Trump appears to respond. The former model said she had an affair with the now-president in 2006 around the time his son Barron was born. "So sad!. Why was the tape so abruptly terminated (cut) while I was presumably saying positive things?"
Cohen's lawyer, Lanny Davis, told the Post that his client didn't make the recordings for any nefarious or untoward purposes.
In April, Donald Trump tweeted that Michael Cohen is a "fine person" who he has "always liked and respected", adding that he didn't see him ever turning on his former boss.