Attorney General William Barr’s expanding review of the Russia probe has evolved into a criminal investigation, according to a person familiar with the matter, giving a federal prosecutor who is leading the inquiry the ability to subpoena witnesses and use a grand jury.

It couldn’t immediately be determined what prompted the change nor what evidence of a crime Mr. Barr and the federal prosecutor leading the probe, John Durham, may have found. When the inquiry was announced in May, Justice Department officials described it...