Khashoggi has not been seen since entering the Saudi consulate building on Tuesday where he was to receive an official document for his marriage.
The BBC said that Khashoggi served as an adviser to the Saudi royal family and was for many years seen as an insider, until he left more than a year ago amid a reported clampdown on press freedom.
It also reported Turkey's Deputy Foreign Minister Yavuz Selim Kiran insisted the issue "should be cleared up immediately".
In a statement on Twitter, the Saudi consulate said it was "following news stories" about Khashoggi and is in co-ordination with Turkish authorities to "uncover the facts behind his going missing after exiting" the consulate building.
A Saudi Press Agency statement said while the consulate did not challenge that Khashoggi disappeared during a visit to the diplomatic post, it was working with Turkish authorities to find the dissident writer.
"According to the information we have, this person who is a Saudi citizen is still at the Saudi Consulate in Istanbul", Kalin said.
Mr Khashoggi, 59, had been living in self-imposed exile in the USA since Mohammed bin Salman, Saudi Arabia's crown prince, came to power in 2017.
Khashoggi's fiancee Hatice, 36, told AFP on Thursday that there had been no news about him. "We don't know where he is". He gave her his mobile phones for safekeeping, something common as embassies throughout the Middle East routinely require phones to be left outside as a security precaution. There's growing concern the columnist, who's a vocal critic of Saudi Arabia's government, may have been detained by Saudi officials.
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The unnamed woman, who is a Turkish citizen, called police when Khashoggi did not emerge at 5 p.m., after the consulate had officially closed.
She said that he was "stressed and sad" that he was forced to go to the building.
A State Department official declined to say whether the two discussed Khashoggi's status, citing "private diplomatic discussions". "We hope that he is safe and that we can hear from him soon".
The Committee to Protect Journalists said the Saudi's disappearance was a cause for alarm "given the Saudi authorities' pattern of quietly detaining critical journalists".
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".
Khashoggi is one of the rare Saudi journalists to raise his voice against repression in the country.
Ties between Turkey and Saudi Arabia have been strained in the past year by a dispute over Qatar.
The source gave no further details but Turkish broadcaster NTV said the ambassador told Turkish officials he had no information about Khashoggi. "It would be unfair and outrageous if he has been detained for his work as a journalist", The Post's global opinions editor, Eli Lopez, said in a statement.