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.