Lawyers for Mr. Trump and Mr. Cohen declined to comment for this article as did American Media.

It is not known how much of the material on Mr. Trump is still in American Media’s possession or whether American Media destroyed any of it after the campaign. Prosecutors have not said whether they have obtained any of the material beyond that which pertains to Ms. McDougal and Ms. Clifford and the discussions about their arrangements.

For the better part of two decades, Mr. Pecker had ordered his staff at American Media to protect Mr. Trump from troublesome stories, in some cases by buying up stories about him and filing them away.

In 2016, he kept his staff from going back through the old Trump tip and story files that dated to before Mr. Pecker became company chairman in 1999, several former staff members said in interviews with The New York Times.

That meant that American Media, the nation’s largest gossip publisher, did not play a role during the election year in vetting a presidential candidacy — Mr. Trump’s — made for the tabloids.

Mr. Pecker also worked with Mr. Trump and Mr. Cohen to buy and bury Ms. McDougal’s story of an affair with Mr. Trump, a practice known as “catch and kill.” Mr. Cohen admitted as much in making his guilty plea last week.

In August 2016, American Media acquired the rights to Ms. McDougal’s story in return for $150,000 and commitments to use its magazines to promote her career as a fitness specialist. But American Media never published her allegations about a relationship with Mr. Trump.

Shortly after American Media completed the arrangement with Ms. McDougal at Mr. Trump’s behest, a troubling question began to nag at Mr. Trump and Mr. Cohen, according to several people who knew about the discussions at the time: What would happen to America Media’s sensitive Trump files if Mr. Pecker were to leave the company?