A woman who says that she and President Trump had an affair over a decade ago offered new details about the alleged relationship and an effort to buy her public silence.

A Friday report published by The New Yorker describes how a tabloid publisher may have moved to “catch and kill” the story and pay off the woman, Karen McDougal, as Mr. Trump’s candidacy gained momentum.

Ms. McDougal, a former Playboy model, wrote an eight-page note, obtained by The New Yorker, describing the relationship, which allegedly began in 2006, while Mr. Trump was married to his current wife, Melania Trump, and lasted about nine months. Ms. McDougal said she regretted signing a contract with American Media Inc., the publisher of The National Enquirer, for the rights to her story.

“I feel let down,” Ms. McDougal told The New Yorker. “I’m the one who took it, so it’s my fault, too. But I didn’t understand the full parameters of it.”