Donald J. Trump and two of his adult children have agreed to be questioned under oath in mid-July by lawyers from the New York State attorney general’s office, unless the state’s highest court intervenes.

The agreement, filed Wednesday in New York State Supreme Court, says that Mr. Trump, Donald Trump Jr. and Ivanka Trump have agreed to appear for testimony that will begin on Friday, July 15, and end the following week.

The questioning will come as the state attorney general, Letitia James, concludes the final phase of her investigation into Mr. Trump and the business practices of his company, The Trump Organization. The agreement follows a number of legal setbacks for the former president, whose lawyers had fought the attorney general for months, hoping to avoid questioning.

Wednesday’s agreement was filed two weeks after a state appeals court ruled to allow the questioning. The court rejected arguments from Mr. Trump’s lawyers that Ms. James’s civil investigation was politically motivated, and that she should be barred from questioning Mr. Trump under oath while he was also under criminal investigation for some of the same business practices.

Alina Habba, a lawyer for Mr. Trump, said soon after that ruling that she would appeal the matter to the Court of Appeals. It is unclear whether the Court will agree to hear the case, but if it does, the three Trump family members may still have a hope of avoiding the interviews.

Another of Mr. Trump’s adult children, Eric Trump, was questioned under oath in October 2020, and invoked his right against self-incrimination in response to more than 500 questions. While Mr. Trump and the two children could decline to answer questions for the same reason, doing so could harm them in Ms. James’s inquiry. In a criminal case, jurors cannot infer anything from a defendant’s refusal to testify, but that does not hold true for civil cases.

Ms. James’s investigation began in March 2019 and has focused on whether Mr. Trump systematically misstated the value of his assets to gain financial advantage with lenders and tax authorities. Because the inquiry is civil, Ms. James cannot file criminal charges, but can file a lawsuit. A lawyer from her office signaled in April such a filing could occur in the near future.

Lawyers from Ms. James’s office are also involved in a criminal investigation being led by the Manhattan district attorney, Alvin L. Bragg. That inquiry had been heading toward an indictment before Mr. Bragg and some of his top aides developed concerns about the strength of the case.

In the civil investigation Mr. Trump was held in contempt of court in April by a state court judge, Arthur F. Engoron, and ordered to pay $10,000 a day until his lawyers filed a number of documents that were sought by the attorney general. The following month, Justice Engoron lifted the contempt fine, but set conditions and said he would reinstate that penalty if Mr. Trump did not comply.

After a hearing involving lawyers for Mr. Trump and Ms. James on Wednesday, the judge said that he would leave the order in place — with no fine attached — and gave Mr. Trump’s lawyers a deadline of next Friday to file information about the Trump Organization’s document retention policies.

In a recent live appearance on the podcast “Pod Save America,” Ms. James said that Mr. Trump “got caught” using “funny numbers in his financial documents.” She also said that she would “allow him to exhaust his appeal.”

Mr. Trump has repeatedly denied wrongdoing and referred to Ms. James’s investigation as a witch hunt and to the attorney general as a radical left “racist.”

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