"He said that they are working closely with Turkey to find answer", Trump wrote.
The kingdom has called such allegations "baseless" but has not offered any evidence Khashoggi left the consulate.
More major players in global finance and industry are pulling out of a high-profile business conference in Saudi Arabia in the wake of reports that a Saudiwriter critical of the government disappeared and is feared dead after entering a Saudi consulate in Turkey.
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.
Prince Khaled al-Faisal, a senior member of Saudi Arabia's ruling family and senior advisor to King Salman, has met Erdogan to discuss Khashoggi's disappearance, two sources with knowledge of the matter told Reuters without providing details.
The Saudi prosecutor has received instructions "to work quickly", so an announcement could happen within days, the Saudi official said.
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.
USA stock futures declined again on Monday as Saudi Arabia faced growing pressure over a missing critic of the kingdom.
The Tadawul exchange in Riyadh dropped by 7 percent at one point during the week's first day of trading, with 182 of its 186 listed stocks showing losses by the early afternoon. Turkish officials say they have proof of his murder and torture at the hands of the Saudis in the form of audio and video taken from Khashoggi's Apple Watch (the Apple Watch portion is nearly certainly a lie made in order to protect Turkish surveillance methods).
"Something this big would definitely spook investors, and Saudi just opened up for foreign direct investment, so that was big", said Issam Kassabieh, a financial analyst at Dubai-based firm Menacorp Finance.
"The kingdom affirms its total rejection of any threats and attempts to undermine it, whether by threatening to impose economic sanctions, using political pressures or repeating false accusations", the statement said.
Britain, France and Germany also released a rare joint statement saying they were treating Khashoggi's disappearance "with the utmost seriousness" and calling for a "credible investigation".
Visitors enter the Saudi Arabia's consulate in Istanbul, Turkey, Monday, Oct. 15, 2018.
The Washington Post, citing unidentified United States and Turkish officials, reported that Turkey had told USA officials it has audio and video recordings that prove Khashoggi was killed inside the consulate.
Salman emphasised the importance of the Turkey-Saudi relationship and said no-one should be able to "undermine the strength of this relationship", it added.
While the kingdom has denied any involvement in Khashoggi, U.S. administration officials are said to increasingly regard the kingdom's position as untenable.
In an interview with CBS" "60 Minutes' broadcast the night prior, Trump said of the situation: "There's a lot at stake..."
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