"We would like to know exactly what happened inside the Saudi Consulate in Istanbul and the circumstances surrounding his disappearance", said Mohamed Okad, a friend of Khashoggi and founder of Insight into Crisis, a conflict advisory group.
Secretary of State Mike Pompeo also said US officials have raised the matter with their Saudi counterparts.
"I am concerned about it", Trump said on the White House South Lawn.
Crown Prince Mohammed bin Salman has previously said officials are welcome to conduct a search as there is nothing to hide. Al-Jazeera, Qatar's state-funded broadcaster, has focused extensively on the case in recent days.
"The consulate officials can not save themselves by simply saying "he has left".
Turkish President Recep Tayyip Erdogan earlier said he was hopeful about the missing Saudi journalist. The consulate has other entrances and exits as well, and Saudi officials insist he left through one of them.
"The ambassador was told that we expected full cooperation during the investigation", the source said.
In Washington, President Donald Trump expressed reservations over withholding American arm sales over the writer, even as prominent American lawmakers increasingly criticize Saudi Arabia - America's longtime security ally in the region.
The suspects reportedly left the area with two cars whose license plates could be read by the police.
The two countries have such an obligation under both criminal law and global human rights law, she said.
Saudi Arabia, while not yet being an official North Atlantic Treaty Organisation member, is a prominent US partner. Relations were already strained after Turkey sent troops to the Gulf state of Qatar previous year in a show of support after its Gulf neighbours, including Saudi Arabia, imposed an embargo on Doha.
Khashoggi went to the consulate to procure paperwork certifying that he had divorced his wife, in order to marry his fiancé, Hatice Cengiz, who is a Turkish national.
The Khashoggi family also released a statement in support of Saudi government efforts to find Jamal: "We trust the government and the actions it has taken and all the efforts being made in the case of Jamal Khashoggi".
Republican Senator Lindsey Graham said Riyadh must provide "honest answers" to the claims that he was murdered.
Mr Khashoggi, who was living in self-imposed exile in the United States and working for the Washington Post before his disappearance, was a frequent critic of the kingdom's de facto ruler Crown Prince Mohammed bin Salman, whom he described as a "brash and abrasive young innovator" who was "acting like Putin".
"If Saudi authorities surreptitiously detained Khashoggi it would be yet another escalation of Crown Prince Muhammad bin Salman's reign of repression against peaceful dissidents and critics", Sarah Leah Whitson, Middle East director at HRW, said.
Journalists and activists gathered outside the Saudi consulate on Monday demanding information on Khashoggi's fate.
Paul acknowledges he and Trump differ on their views of Saudi Arabia.
A Turkish official also told The Associated Press that detectives' "initial assessment" was that Khashoggi was killed at the consulate, without elaborating.
The Washington Post chose to leave a blank space where Khashoggi's column would have been in its Friday edition in support of the missing writer.
Khashoggi has been a vociferous critic of Prince Mohammed's policies in both the Arab and Western press.