The company that owns the National Enquirer, a backer of Donald Trump, agreed to pay $150,000 to a former Playboy centerfold model for her story of an affair a decade ago with the Republican presidential nominee, but then didn’t publish it, according to documents reviewed by The Wall Street Journal and people familiar with the matter.

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.

Hope Hicks, a Trump campaign spokeswoman, said of the agreement with Ms. McDougal: “We have no knowledge of any of this.” She said that Ms. McDougal’s claim of an affair with Mr. Trump was “totally untrue.”

Ms. McDougal expected her story about Mr. Trump to be published, people familiar with the matter said. American Media didn’t intend to run it, said another person familiar with the matter. Ms. McDougal didn’t return calls for comment.