Even if the Manhattan district attorney’s office is successful in obtaining the president’s tax returns, the documents would be covered by secrecy rules governing grand juries, meaning they would not become public unless they were used as evidence in a criminal case.
At the beginning of August, the state prosecutors also subpoenaed the Trump Organization, seeking documents related to the payment to Ms. Daniels and the reimbursement to Mr. Cohen. With few legal options, the Trump Organization has been complying with that subpoena.
Still, the company has derided the investigation by Mr. Vance, a Democrat, as politically motivated.
“It’s just harassment of the president, his family and his business, using subpoenas as weapons,” Mr. Mukasey said last month.
As part of its investigation, prosecutors from Mr. Vance’s office visited Mr. Cohen in prison in Otisville, N.Y., to seek assistance with their investigation, according to people briefed on the meeting, which was first reported by CNN.
Mr. Cohen also helped arrange for American Media Inc., the publisher of The National Enquirer, to pay Karen McDougal, a Playboy model who also said she had an affair with the president. Prosecutors in the district attorney’s office subpoenaed American Media in early August, as well as at least one bank.
The investigation is not the first time Mr. Vance’s office has focused on members of the Trump family or its business. In 2012, his office declined to charge two of Mr. Trump’s children, Ivanka Trump and Donald Trump Jr., in an investigation into whether they misled buyers interested in the Trump SoHo hotel-condominium project, a decision that resulted in criticism of Mr. Vance.
Maggie Haberman contributed reporting.
