The deal and the extent of Mr. Cohen’s role in it are the subjects of a wide-ranging federal corruption investigation that is, in part, looking into his efforts to protect Mr. Trump’s presidential prospects in 2016.

A.M.I. indicated earlier this month that it would fight Ms. McDougal, asking the Los Angeles Superior Court to dismiss her lawsuit.

But that was roughly a week before federal investigators obtained email communications, audio recordings and other documentation from Mr. Cohen during their raid of his office, home and hotel room. Those materials included information about A.M.I. and the McDougal suit, people involved in the case said.

The suit also claimed that Mr. Cohen had been secretly involved in the talks between A.M.I. and Ms. McDougal’s lawyer at the time, Keith M. Davidson — who emailed Mr. Cohen at the end of the negotiations. A spokesman for Mr. Davidson has said the lawyer “fulfilled his obligations and zealously advocated for Ms. McDougal.”

A.M.I. also spoke with Mr. Cohen about Ms. McDougal, though it says it did so only as part of its reporting process.

Mr. Stris said that before reaching the settlement, he was prepared to answer A.M.I.’s motion to dismiss Ms. McDougal’s case with a request for a limited version of pretrial discovery. The move, compelling both sides to share emails and other records, could have provided information that would not be available through the material the F.B.I. seized from Mr. Cohen. An A.M.I. lawyer, Cameron Stracher, said that he doubted the request would have succeeded, and that avoiding discovery was not a motivating factor in the settlement.

The agreement precludes any of that from happening, at least in Ms. McDougal’s civil case against A.M.I., though Mr. Stris said he expected federal investigators to eventually secure everything they needed to fully vet the process behind the deal. “I have tremendous confidence in the men and women of the Southern District of New York,” he said, referring to the federal prosecutors investigating Mr. Cohen.