The Trumps have until June 13 to petition the highest court in New York state, the Court of Appeals, to further delay any testimony in Ms. James’ investigation.

Letitia James, state attorney general, is examining whether the Trump Organization misled lenders and tax officials.

The agreement was reached more than six months after Ms. James issued the initial subpoenas for their testimony, which Mr. Trump, 75, and his children, both in their forties, fought vehemently in court.

A state court of intermediate appeals refused Trump’s request to avoid testifying last month.

The Trumps maintained that their constitutional rights would be violated if they testified in the civil investigation because their remarks may be utilized in a parallel criminal investigation led by Manhattan district attorney Alvin Bragg.

According to an agreement with Ms. James made public on Wednesday, their testimony might begin on July 15 and run through the following week.

Allen Weisselberg, the longtime top financial officer of Trump’s family business, was charged with tax evasion in a Manhattan investigation last year.

He has entered a not-guilty plea and is scheduled to stand trial later this year.

Republican Mr. Trump has denied any wrongdoing and stated that the probe is motivated by partisan politics since Ms. James is a Democrat.

A federal court in the state capital of Albany denied Mr. Trump’s separate lawsuit attempting to halt the investigation because she was politically biassed last month.

The judge found no indication of bad faith on her part.

The Trumps have until June 13 to petition the highest court in New York state, the Court of Appeals, to further delay any testimony in Ms. James’s investigation.