"The consulate officials can not save themselves by simply saying, 'He has left, '" Erdogan said Monday on a visit to Budapest.
"Hopefully that will sort itself out", he added.
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.
Police also said a team of 15 Saudis were sent to Istanbul and were in the building at the same time as Khashoggi.
Turkey has said it will search the consulate for signs of what happened to Khashoggi, but has not said when it will do so.
Turkey summoned the Saudi ambassador on Sunday to request the kingdom's "full cooperation" in the investigation, a Foreign Ministry official said. "We are aware of the latest reports and are working urgently to establish the facts, including with the government of Saudi Arabia".
"Just spoke to Senators Bob Corker and Ben Cardin about our shared concerns regarding the whereabouts and treatment of Mr Jamal Khashoggi", Republican Senator Lindsey Graham tweeted Monday.
Saudi Arabia calls the accusation "baseless", but has also offered nothing to prove he ever left the building.
Saudi Arabia's Ambassador to the United States Prince Khalid Bin Salman has responded to what he termed as "grim and malicious" rumors over the investigation surrounding the disappearance of Saudi journalist Jamal Khashoggi.
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Turkish authorities have told MEE they believe Khashoggi, a prominent journalist and critic of the Saudi govenrment, was killed soon after entering the building.
The Washington Post, for which the self-exiled Khashoggi wrote columns over the past year, pressed for release of evidence.
Khashoggi hadn't lived in Saudi Arabia since he moved to the United States past year. If Mr. Khashoggi was murdered inside the consulate, it will cast the Saudi regime and its de facto ruler, Crown Prince Mohammed bin Salman, in a new and disturbing light and require a thorough reevaluation of U.S. -Saudi relations.
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". "Right now, nobody knows anything about it, but there are some pretty bad stories going around".
Khashoggi was at the consulate to obtain paperwork so he could marry his Turkish fiancée, Hatice Cengiz.
"There is a video of the moment of him being killed", Kemal Ozturk, a columnist in a pro-government newspaper and the former head of a semiofficial news agency, said in an interview on a pro-government television network, citing unnamed security officials.
Trump, who has long criticized the USA media, on Monday said he was concerned about reports regarding the journalist and did not "like hearing about it", but that he hoped the situation "will sort itself out".
The newspaper said that two private Gulfstream IV planes belonging to a charter service that has often worked with the Saudi government, Sky Prime Aviation, took off from Riyadh on the morning of October 2 carrying 15 passengers - all intelligence agents or government officials.
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