The consulate said it was "following up reports of the disappearance" of Jamal Khashoggi, who went there to complete paperwork on Tuesday.
Khashoggi's fiancée accompanied him to the building, but she had to wait outside. She did not see him leave.
Turkish authorities have also said they believe Khashoggi is still there.
The Turkish foreign ministry summoned Saudi Arabia's ambassador on Wednesday and "asked for an explanation" about the disappearance, NTV television reported.
The ambassador informed Deputy Foreign Minister Yavuz Selim Kiran that he had no information about Khashoggi's whereabouts, but Mr Kiran told him that the issue "should be cleared up immediately", it added.
The US state department has also requested information about Khashoggi's whereabouts and expressed concern about his safety.
The 59-year-old journalist had been living in self-imposed exile in the US since last year, when he left the Gulf kingdom after criticising the policies of Saudi Crown Prince Mohammed bin Salman.
He is a regular contributor to the Washington Post, which said it had "reached out to anyone we think might be able to help locate him".
What happened on Tuesday?
Khashoggi went to the consulate to obtain a document certifying he had divorced his ex-wife, so that he could marry his Turkish fiancée, Hatice.
She said that he was "stressed and sad" that he was forced to go to the building.
Khashoggi was required to surrender his mobile phone, which is standard practice in some diplomatic missions. Hatice said he left the phone with her and told her to call an adviser to Turkish President Recep Tayyip Erdogan if he did not return.
Hatice said she waited for Khashoggi outside the consulate from about 13:00 (10:00 GMT) until after midnight and did not see him leave. She returned when the consulate reopened on Wednesday morning.
What does Saudi Arabia say?
A Saudi official confirmed on Wednesday that Khashoggi had visited the consulate to complete paperwork, but said he had "exited shortly thereafter".
"He is not in the consulate nor in Saudi custody," the official added.
On Thursday, the official Saudi Press Agency cited the consulate as saying it was "carrying out follow-up procedures and co-ordination with the Turkish local authorities to uncover the circumstances of the disappearance of Jamal Khashoggi after he left the consulate building".
What does Turkey think?
Turkish presidential spokesman Ibrahim Kalin told reporters on Wednesday: "According to the information we have, this person who is a Saudi citizen is still at the Saudi consulate in Istanbul."
"We don't have information to the contrary," he added.
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