According to the Washington Post, Ms Cengiz said Khashoggi was concerned he would not be allowed to leave - before entering the consulate, he gave her his phone and instructions to call a Turkish official if he did not return.
Turkish police were limited to footage from two street cameras that monitor the street outside the consulate after Saudi officials claimed that security cameras inside the consulate were not working at the time of last week's incident.
"Saudi authorities said they were open to cooperation and that a search can be conducted at the consulate building", the Turkish foreign ministry spokesman Hami Aksoy said in a statement.
He added that investigations are still ongoing. For his part, Trump simply said, "Hopefully that will sort itself out".
Last week, Crown Prince Mohammed told Bloomberg News that his government was "very keen to know what happened to him", and that Mr Khashoggi had left "after a few minutes or one hour".
Mr Hunt's intervention followed the publication of a CCTV image apparently showing Mr Khashoggi walking into the Saudi consulate in Istanbul, just before he went missing.
But if Khashoggi is dead, the Saudi government certainly had a motive to kill him.
A report Monday in the daily newspaper Sabah said investigators were also focused on a convoy of diplomatic vehicles that departed from the consulate on the day Khashoggi vanished. "It would be a violation of worldwide law to harm, arrest or detain people at a diplomatic mission, he said, and noted that no such thing had ever happened in Turkey's history".
Ousted Guardians of the Galaxy director switches camps, tied to new DC Comics film
Fresh off his Marvel firing, James Gun is reportedly in talks to write and potentially direct DC's Suicide Squad sequel. O'Connor had been hired to both write and direct the film and was hard at work on the project for the past year.
Crown Prince Mohammed's brother and the Saudi ambassador to the US, Khaled, has insisted all the reports about his disappearance or death "are completely false and baseless".
"If he left, you must prove this, you will prove this, even if it's with visuals".
Turkish officials have said he was murdered there, while Saudi Arabia says he safely left the building.
Turkish officials told Reuters news agency over the weekend that they believed Khashoggi had been killed inside the consulate, allegations Saudi Arabia rejects. As a columnist for the Post, Khashoggi was a frequent critic of the Saudi regime.
Saudi Arabia has denied involvement.
Mr Khashoggi went to the consulate last Tuesday to obtain a document certifying he had divorced his ex-wife, so that he could remarry his Turkish fiancée, Hatice Cengiz. "Right now, nobody knows anything about it, but there are some pretty bad stories going around".
The Saudi writer spent previous year in the U.S.in self-imposed exile, after he fled the kingdom amid a crackdown on intellectuals and activists who criticized the policies of Crown Prince Mohammed bin Salman.
But Khashoggi's fiancée is reportedly holding out hope, tweeting in Arabic late Saturday and translated by the Middle East Eye, "Jamal was not killed, and I don't believe that he was killed", the fiancée said.
Accusations the Saudi leadership directly ordered the alleged assassination of Khashoggi will put further pressure on the United States and other allies to demand a transparent investigation, with possible serious repercussions to bilateral relations if it doesn't come to fruition. Why don't you prove it? "But if this deeply disturbing news report is confirmed, the United States & the civilized world must respond strongly, and I will review all options in Senate".