The tabloid-newspaper publisher reached an agreement in early August with Karen McDougal, the 1998 Playmate of the Year. American Media Inc., which owns the Enquirer, hasn’t published anything about what she has told friends was a consensual romantic relationship she had with Mr. Trump in 2006. At the time, Mr. Trump was married to his current wife, Melania.

Quashing stories that way is known in the tabloid world as “catch and kill.”

In a written statement, the company said it wasn’t buying Ms. McDougal’s story for $150,000, but rather two years’ worth of her fitness columns and magazine covers as well as exclusive life rights to any relationship she has had with a then-married man. “AMI has not paid people to kill damaging stories about Mr. Trump,” the statement said…

A contract reviewed by the Journal gave American Media exclusive rights to Ms. McDougal’s story forever, but didn’t obligate the company to publish it and allowed the company to transfer those rights. It barred her from telling her story elsewhere. The company said it also would give her monthly columns to write and would put her on magazine covers.