Turkish officials have said they believe the Saudi journalist was murdered there and his body removed, which the Saudis have strongly denied.
The president provided no evidence to back his comment, but said that during a phone call the Saudi King had firmly denied knowing what had happened to Mr Khashoggi.
The consulate was the last place Khashoggi was seen before he vanished on October 2. His bride, a citizen of Turkey, which came with him to the building, but left to wait for him on the street, said that from the Consulate, he has failed.
Khashoggi, a Washington Post columnist and USA resident, has not been seen since entering the consulate on October 2 to obtain documents for his forthcoming marriage.
Lurid claims have appeared in Turkish media, including that Khashoggi was tortured and dismembered.
Diplomatic pressure is growing on the Saudis to give a fuller explanation.
Trump dispatched Pompeo shortly after the Monday call with King Salman.
Trump said King Salman added that Crown Prince Mohammed bin Salman also "had no knowledge" about Khashoggi's fate.
The official said the prosecutor had been instructed to work quickly.
On Sunday, Saudi Arabia warned that if it "receives any action, it will respond with greater action, and that the kingdom's economy has an influential and vital role in the global economy".
How was the consulate search conducted?
Shortly after the cleaning crew entered the Saudi consulate, a team of investigators arrived in an unmarked police vehicle to begin an inspection of the building a full 13 days after Khashoggi entered to obtain marriage documents and never reemerged.
"It's important to establish clear facts about what has happened, and it's important for the worldwide community to be clear that those facts need to be established in a clear and transparent manner". He is embroiled in a damaging dispute with neighbouring Qatar.
CCTV Footage of Jamal Khashoggi entering the Saudi Embassy in Turkey. But he left Saudi Arabia roughly a year ago over fears for his safety after writing articles critical of the government, particularly Crown Prince Mohammed bin Salman.
Trump's interview ー in which he opined on several issues in confident and defiant terms, at one point telling CBS's ($CBS) Lesley Stahl, "I'm president and you're not" ー is part of a coordinated media strategy that has allowed the president to hold multiple Q&As per day.
Turkish authorities have said he was killed at the consulate.
The prince, ambitious, aggressive and just 33 in a kingdom long ruled by aging monarchs, has considerable weight in Saudi government actions.
What other reaction has there been?
BlackRock, the world's largest fund manager, and Blackstone, the world's biggest manager of alternative investments had both previously planned to attend the Future Investment Initiative conference in Riyadh.
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