Earlier this year, Mr. Trump firmly told reporters aboard Air Force One that he had not known about the payments. But on an audio recording made by Mr. Cohen, and seized during an F.B.I. raid in April, Mr. Trump could be heard talking to Mr. Cohen about payments to executives at American Media Inc., the parent company of the National Enquirer, to acquire years of dirt about the candidate.

Prosecutors also revealed in a sentencing submission last week that Mr. Trump attended at least one meeting involving Mr. Cohen and David Pecker, the A.M.I. chairman, during which Mr. Pecker “offered to help deal with negative stories” about Mr. Trump’s affairs with women “by identifying such stories so that they could be purchased and ‘killed.’”

The government also suggested that in addition to Mr. Trump, others associated with the Trump campaign may have been involved, and said that Mr. Cohen “coordinated his actions with one or more members of the campaign, including through meetings and phone calls, about the fact, nature and timing of the payments.”

Prosecutors have not given any indication that criminal charges against the president would be forthcoming, particularly as the Justice Department has long taken the position that sitting presidents are not subject to criminal prosecution.

Norman L. Eisen, the co-founder of Citizens for Responsibility and Ethics in Washington, said prosecutors in the Southern District of New York appeared to be building a broader case against the Trump Organization and the Trump campaign, possibly for whatever role they may have played regarding illegal campaign donations, using Mr. Cohen’s guilty plea as a steppingstone.

“The big fish here are the Trump campaign, the Trump Organization and, as an unindicted co-conspirator, the president himself,” Mr. Eisen said.

For months, the president has demeaned Mr. Cohen, calling him weak and praising other former aides who have refused to cooperate with the authorities. Mr. Trump also suggested on Thursday that, despite the plea deal by Mr. Cohen, his family could face future charges — something prosecutors have not raised publicly.