Michael Cohen recorded a conversation he had with a reporter this year in which he said he arranged “on my own” a $130,000 payment in 2016 to a former adult-film star who alleged an affair with Donald Trump, according to people familiar with the matter.

In the nearly two-hour conversation with CNN reporter Chris Cuomo, which the people said appears to have been surreptitiously recorded by Mr. Cohen, the former Trump lawyer discussed at length the payment he arranged in October 2016, a month before the presidential election, to Stephanie Clifford, known professionally as Stormy Daniels.

“I did it on my own,” Mr. Cohen said of the payment, according to the people familiar with the matter.

Mr. Cohen’s claims of personal responsibility for the deal echoed his public comments after The Wall Street Journal in January revealed that Mr. Cohen had arranged the payment to Ms. Clifford. Mr. Cohen said in a February statement that he “facilitated” the payment using his own funds, calling it a “private transaction.”

Mr. Trump’s lawyer, Rudy Giuliani, later disclosed that Mr. Trump reimbursed Mr. Cohen for the payment through a monthly retainer but hadn’t learned of the deal with Ms. Clifford until afterward. The White House could use the tape to try distance Mr. Trump from the deal or to impeach Mr. Cohen’s credibility if he decides to cooperate with prosecutors and implicates Mr. Trump in the deal.