Riyadh says Jamal Khashoggi, a contributor to the Washington Post, had left the consulate but Ankara says he is still inside.
The news agency quoted Istanbul prosecutors as saying that an investigation had been launched on Tuesday and had been widened since.
Khashoggi went to the consulate to receive an official document for his marriage.
A former government adviser who has been critical of some policies of Saudi Crown Prince Mohammed bin Salman and Riyadh's intervention in the war in Yemen, Khashoggi has lived in self-imposed exile in the United States since last year to avoid possible arrest.
The Saudi crown prince has told Bloomberg that the journalist was not inside the consulate and said he was ready to allow Turkish authorities to search the building.
"We are ready to welcome the Turkish government to go and search our premises," he said, which is Saudi sovereign territory.
He added "we will allow them to enter and search and do whatever they want to do... We have nothing to hide," he said in the interview published on Friday.
According to Khashoggi's fiancee, a Turkish woman identified only as Hatice A., he went to the consulate and never re-emerged.
Ankara and Riyadh have given contradictory versions of the circumstances of Khashoggi's disappearance.
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