Mr Khashoggi was last seen at the Saudi consulate in Istanbul two weeks ago.
Turkish officials believe Mr Khashoggi was murdered by Saudi agents but the Saudis have denied this.
However, US media are reporting that the Saudis may be preparing to admit that Mr Khashoggi died as a result of an interrogation that went wrong.
Overnight, Turkish police completed a search of the consulate after being admitted by Saudi authorities.
What’s likely to come from the Pompeo meeting?
The secretary of state and the king have now met in Riyadh. There is no official word yet on the meeting, but Mr Pompeo would have been looking for the king to expand on his conversation with President Donald Trump on Monday.
Tweeting about the call, Mr Trump said: “Just spoke to the king of Saudi Arabia who denies any knowledge of whatever may have happened ‘to our Saudi Arabian citizen’.”
He later told reporters: “The denial was very, very strong. It sounded to me like maybe these could have been rogue killers. Who knows?”
There is a lot at stake given the strength of Saudi-US ties. Mr Trump has already ruled out cancelling a lucrative arms deal, although he did threaten “severe punishment” if the kingdom were found to be responsible for the death.
King Salman ordered an investigation into the missing journalist on Monday. Saudi statements up to now have dismissed allegations of a killing as “baseless” and “lies”.
The kingdom has also angrily rejected what it called political and economic “threats”, saying it would respond to any punitive action, such as sanctions, “with a bigger one”.
Mr Pompeo is also expected to meet Crown Prince Mohammed bin Salman during his day in Riyadh. The secretary of state may then head to Turkey.
Reflecting the growing pressure on Saudi Arabia, UN High Commissioner for Human Rights Michelle Bachelet on Tuesday demanded that the kingdom and Turkey reveal all that they know about the case.
She said that the diplomatic inviolability of the consulate and the diplomatic immunity of officials should be waived during the investigation.
What is this US media line about?
It appeared in the New York Times and on CNN, quoting unnamed sources.
They said Saudi Arabia would acknowledge that Mr Khashoggi’s death was the result of an interrogation that went wrong and the intention had been only to abduct him from Turkey.
This may explain in part Mr Trump’s “rogue killers” line.
Who such killers could be and how it fits into reports of a Saudi team being despatched to the consulate before Mr Khashoggi’s arrival will presumably need to covered.
CNN said the Saudis may argue the operation was conducted without clearance and those responsible would be held accountable.
The Khashoggi family in Saudi Arabia issued a statement calling for an “independent and impartial international commission”.
What happened with the consulate search?
For the first time since the journalist disappeared on 2 October, Turkish investigators were allowed to enter the building.
A Saudi team entered first on Monday, followed roughly an hour later by Turkish forensic police.
The Turkish investigators, some wearing overalls, gloves and covered shoes. stayed for about eight hours, leaving in the early hours of Tuesday.
They reportedly took with them samples, including of soil from the consulate garden.
Saudi Arabia agreed last week to allow Turkish officials to conduct a search but insisted it would only be a superficial “visual” inspection.
Turkey rejected that offer. The Sabah daily newspaper said investigators had wanted to search the building with luminol, a chemical which shows up any traces of blood. It is not clear whether that happened.
Reports on Tuesday said Istanbul police would also search the Saudi consul’s residence.
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