Judge Wood made it clear that Ms. Jones was well positioned to serve as an arbiter. For 17 years, she served on the bench in the same courthouse where the Cohen case is now being heard. Before that, she worked as a prosecutor in the United States attorney’s office for the Southern District of New York. Ms. Jones was also once a senior aide to Robert M. Morgenthau, the former Manhattan district attorney.

As a private lawyer, Ms. Jones has worked as a monitor or a “neutral party” in disputes. In 2014, she was picked to hear the appeal by Ray Rice, a former Baltimore Ravens player, of the National Football League’s decision to suspend him over a domestic-abuse case.

In the past week or so, Mr. Cohen has been shedding legal exposure in separate but related court cases. On Wednesday, he filed papers in California saying he planned to invoke his Fifth Amendment right against self-incrimination in a lawsuit filed last month against President Trump by Stephanie Clifford, a former pornographic film star better known as Stormy Daniels, who claims she had an affair with the president.

Mr. Cohen has admitted to paying Ms. Clifford $130,000 in exchange for signing a nondisclosure agreement in October 2016, a month before the election. Ms. Clifford’s suit seeks to invalidate the agreement, alleging it was void because Mr. Trump never put his name to it.

Mr. Cohen also dropped two defamation lawsuits last week against the political research firm Fusion GPS and the website BuzzFeed related to their publication of the so-called Steele dossier, an intelligence report alleging ties between the Trump campaign and Russia. In the suits, Mr. Cohen had accused both companies of disseminating the dossier even though both knew that it contained what he described as lies.

All three moves may help Mr. Cohen avoid releasing documents or providing answers to probing questions that could have been used against him in the criminal investigation. That inquiry is said to be focusing not only on hush-money payments that Mr. Cohen made to Ms. Clifford, but also on his role in keeping quiet another woman who claims she had an affair with Mr. Trump, Karen McDougal, a former Playboy model.

On Thursday, Ms. Clifford’s lawyer, Michael Avenatti, filed a motion on her behalf to formally intervene in Mr. Cohen’s case in New York.