Britain, a close Saudi ally, said the incident could affect relations with Riyadh if they are behind the incident.
Saudi Arabia has described the allegation as unfounded, but has offered no proof that Khashoggi left the consulate alive, nor has Turkey produced evidence that he was killed inside the diplomatic outpost.
Local media said Ankara requested permission for Turkish investigators to search the consulate building, but a Foreign Ministry official would not confirm the report. "And - but let me just say these are senior Foreign Service officers that have had full careers, and we're confident in our team's ability", he said.
Khashoggi's fiancé told the authorities that he was required to submit his cell phone to Saudi officials upon entering the building - a practice that is not uncommon.
Trump had spoken about Khashoggi's disappearance briefly with reporters on Monday.
A demonstrator dressed as Saudi Arabian Crown Prince Mohammed (centre) with blood on his hands protests with others outside the Saudi Embassy in Washington, DC, demanding justice for missing Saudi journalist Jamal Khashoggi.
The image shows the date and time (13:14) and bears a caption in Turkish with the journalist's name and that he was arriving at the consulate.
Saudi authorities have said they do not have security footage from that day.
Saudi officials insist he left through one of the consulate's doors.
However, a number of closed-circuit surveillance cameras surround the area.
Spokeswoman Sarah Huckabee Sanders said the U.S.is continuing to monitor the unfolding investigation in Istanbul, but offered no information about what the crown prince told the USA officials about Khashoggi's disappearance. However, they have failed to provide any proof that the journalist left.
Khashoggi had gone to the consulate in Istanbul for paperwork to marry his Turkish fiancée.
Turkish President Recep Tayyip Erdogan, in reference to the case, said on Monday that Saudi officials have the responsibility to prove the whereabouts of the missing writer. "We can not remain silent in the face of an event like this because it is not a normal occurrence", Erdogan said while returning on a flight from Hungary. "Why don't you prove it - you have to prove it". We're also joined by Medea Benjamin, co-founder of CodePink, who knew Khashoggi and is the author of "Kingdom of the Unjust: Behind the U.S. -Saudi Connection". Turkey sent food to Qatar and deployed troops at its military base there.
Erdogan himself has not publicly accused Saudi Arabia of killing Khashoggi.
The prince promised social and economic reform, but Khashoggi pointed to the escalating crackdown on dissent voices and the media in Saudi Arabia.
The U.S. State Department and President Donald Trump say they're concerned about the disappearance of Saudi dissident journalist Jamal Khashoggi, but they have no idea what happened to him.