Trump dispatched Secretary of State Mike Pompeo for what the State Department described as "face to face meetings with the Saudi leadership" to find out what happened.
On Tuesday, Turkish investigators searched the consulate building.
Saudi officials have called Turkish allegations that the kingdom killed Khashoggi "baseless", but reports in U.S. media on Tuesday (Wednesday NZT) suggested the Saudis may acknowledge the writer was killed at the consulate, perhaps as part of a botched interrogation.
The Turkish foreign minister, Mevlüt Çavuşoğlu, will have briefed him on the Turkish evidence that Saudi intelligence agents, including anaesthetists and some carrying hacksaws and other devices, arrived in the country to conduct a violent interrogation meant to permanently silence the crown prince's most persistent and eloquent critic.
The crown prince also agreed on the need for an investigation that "provides answers", she said.
On Monday, however, Trump offered a different theory after speaking by telephone with King Salman.
"He didn't really know, maybe - I don't want to get into his mind but it sounded to me - maybe these could have been rogue killers".
"This is going to alter the relationship between the U.S. and Saudi Arabia for the foreseeable future", Sen.
Mr Pompeo is expected in Turkey later. Both countries' leaders know they have an enormous amount to lose if this affair ends up splitting apart their 73-year-old strategic partnership.
Saudi Arabia's Public Investment Fund, which hosts the conference, has tentatively committed $20 billion to an infrastructure investment planned with Blackstone Group.
Officials said the dilemma comes at a fraught moment as the USA is expected to reimpose harsh sanctions against Iran on November 5, with the intent of cutting off all Iranian oil exports.
The senior Turkish officials said they were eager to interview members of the 15-man Saudi team, all of whom were believed to have arrived and left Istanbul on the day Khashoggi disappeared.
While campaigning in 2015, for instance, he boasted about his business dealings with the Saudis, saying, "Saudi Arabia, I get along with all of them". Certain areas of the consulate were to remain off-limits, although officials would be able to inspect surveillance cameras, Turkish media reported.
Fox Business Only Major Press to Back Saudi Conference
After committing $45 billion to the Vision Fund, Saudi Arabia's crown prince Mohammed bin Salman also pledged to invest billions in SoftBank's next venture capital fund.
Some reports have indicated the Saudi regime may soon acknowledge that Khashoggi died in an interrogation gone wrong.
Yesterday, Trump appeared to preview the Saudi line that "rogue killers" could be responsible for Khashoggi's death. Publicly, Saudi Arabia so far has issued only denials in response to reports of Turkish claims that Khashoggi was murdered.
Despite Western concerns about Saudi Arabia's human rights record, Trump still says he is unwilling to pull out of weapons sales agreements with Riyadh.
The search of the consulate continued into the early hours of Tuesday.
On the day of Mr Khashoggi's disappearance, several vehicles with Saudi diplomatic number plates were seen on video footage moving from the consulate to the residence.
Hours before the investigators entered the consulate, multiple news outlets reported of, and cameras filmed, a cleaning crew wheeling in boxes and boxes of cleaning supplies. Permission apparently came after a late Sunday night call between King Salman and Turkish President Recep Tayyip Erdogan.
Mr Erdogan said: "The investigation is looking into many things such as toxic materials and those materials being removed by painting them over".
What allegedly happened in Istanbul?
Khashoggi went missing earlier this month at the Saudi consulate in Turkey.
Khashoggi, a familiar face on Arab talk shows, moved to the United States previous year fearing retribution for his criticism of Prince Mohammed, who has cracked down on dissent with arrests.
There have been widespread reports that Khashoggi was killed inside the consulate by a Saudi hit squad that may have dismembered his body.
The Saudi consulate referred Reuters to authorities in Riyadh, who did not respond to questions about the 15 Saudis.