Turkish officials say they fear a Saudi hit team killed Khashoggi, who has been missing almost two weeks.
Turkish and Saudi authorities have now been told to launch a "credible investigation" into Mr Khashoggi's disappearance.
And President Donald Trump said he does not want to jeopardize a major arms sale deal to Saudi Arabia that he said was worth $110 billion, though defense experts have disputed that amount and have said it's hard to calculate precisely how many jobs would be at stake. Saudi Arabia, meanwhile, said King Salman thanked Erdogan "for welcoming the kingdom's proposal" for forming the working group.
Khashoggi has written extensively for the Post about Saudi Arabia, criticizing its war in Yemen, its recent diplomatic spat with Canada and its arrest of women's rights activists after the lifting of a ban on women driving.
A man enters Saudi Arabia's consulate on October 12, 2018 in Istanbul, Turkey.
On Sunday, the Saudi Embassy followed up by writing on Twitter: "The Kingdom of Saudi Arabia extends...appreciation to all, including the United States administration, for refraining from jumping to conclusions on the ongoing investigation".
The Saudi government has described the allegations as "baseless", but have offered no evidence that Mr Khashoggi left the consulate.
The Journal, like CNN, said the Saudi statement has not been finalized.
"I have asked Secretary of State Mike Pompeo to get on a plane and go to Saudi Arabia. go to Turkey if necessary".
Until now, Riyadh has not allowed Turkish investigators to search the consulate - officially Saudi territory - with reports both sides were at odds over the conditions.
Yet Meyer says Saudi Arabia's threats are a "bluff" that amount to more of a tactic to save face in front of a domestic audience.
The Saudis have insisted that Khashoggi, who would have been 60 on Saturday, left the building...
A Saudi official, not authorised to speak publicly, told Reuters that the king had ordered an internal investigation based on information from the joint team in Istanbul.
Saudi Arabia has responded to Western statements by saying it would retaliate against any pressure or economic sanctions "with greater action", and Arab allies rallied to support it, setting up a potential showdown between the world's top oil exporter and its main Western allies.
Late Sunday, King Salman spoke on the telephone with President Erdoğan about Khashoggi.
Uber's chief executive Dara Khosrowshahi said he will not be attending the event in Riyadh due to his concern over Mr Khashoggi's unknown fate.
Treasury Secretary Steven Mnuchin still plans to attend a previously scheduled Saudi conference this week to address terrorist financing, but those plans could change as the investigation progresses, said White House economic adviser Larry Kudlow. Several business leaders and media outlets have backed out of the upcoming investment conference in Riyadh, called the Future Investment Initiative.