After several of his friends were arrested, his column was cancelled by the al-Hayat newspaper and he was allegedly warned to stop tweeting, Mr Khashoggi left Saudi Arabia for the United States, from where he wrote opinion pieces for the Washington Post and continued to appear on Arab and Western TV channels. Turkish officials have said they fear the columnist was killed at the premises.
Officials say the convoy left the consulate around two hours after Khashoggi entered.
Asked about the case while speaking with reporters outside the White House on Monday, Trump replied, "I am concerned about it".
Secretary of State Mike Pompeo also said US officials have raised the matter with their Saudi counterparts.
Trump has instead focused on U.S. and Saudi shared interests in ratcheting up pressure on Iran. "It has to prove that Jamal wasn't oppressed at the consulate and that he left safely". It has allowed journalists into the consulate to show Mr Khashoggi is not there, reportedly even opening cupboards.
Turkish investigators have hinted they know more about the disappearance than they have disclosed.
The Sabah newspaper published images of the men apparently taken at passport control.
The Saudis have called the allegations "baseless", insisting Khashoggi left the building through an exit that wasn't the main building. Security camera footage shows boxes being loaded into the van, which carried diplomatic number plates. Closed-circuit TV camera feeds from the preschool opposite the rear entrance have been retrieved by Turkish intelligence, and images from outside the hotels are also being reviewed, according to people familiar with the probe.
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A Turkish official told the Reuters news agency, "The initial assessment of the Turkish police is that Mr. Khashoggi has been killed at the consulate of Saudi Arabia in Istanbul".
But he added that diplomatic cars had been seen moving in and out. He has not been heard from since October 2, when he entered a Saudi Arabian consulate in Turkey to receive marriage paperwork.
Earlier on Tuesday, foreign secretary Jeremy Hunt met the Saudi ambassador, Prince Mohammed bin Nawaf Al Saud, to voice the UK's concern about the disappearance Mr Khashoggi.
The worst overall mainstream offender, unsurprisingly, is New York Times opinion writer Thomas Friedman, who had been the Crown Prince's biggest cheerleader.
The foreign secretary, Jeremy Hunt, has warned Saudi Arabia that its friendship with the United Kingdom depends on shared values, after he demanded answers over claims that a Saudi hit squad may have played a role in the disappearance and possible killing of Jamal Khashoggi. Why don't you prove it?
Turkey summoned the Saudi ambassador on Sunday to request the kingdom's "full co-operation" in the investigation, a Foreign Ministry official said. Turkey sent food to Qatar and deployed troops at its military base there as the other nations imposed a boycott on the wealthy Gulf country.
The Committee to Protect Journalists (CPJ) has been documenting the increasingly harsh treatment of journalists in Saudi Arabia.
Khashoggi's disappearance could put pressure on the Saudi prince, who has promoted an image of himself as a reformer and a reliable Western ally.
Trump's son-in-law Jared Kushner maintains close personal ties with the crown prince.