"We hope to have results very quickly", he added.
Turkish officials told Reuters over the weekend that they believed Khashoggi had been killed inside the Saudi consulate in Istanbul, and President Tayyip Erdogan said he was personally following the case.
"If Khashoggi was indeed murdered inside a diplomatic facility, it is an act of terror that echoes Russian and Chinese tactics of extraterritorial, extrajudicial attacks on dissidents, meant to intimidate any who would speak out against the Saudi government, no matter where they may be, and giving the lie to official narratives of "reform" in Saudi Arabia", said Summer Lopez, PEN America's senior director of free expression programs.
The Post cited one anonymous official who said investigators believe a 15-member team "came from Saudi Arabia".
On Monday, a Turkish official also said Saudi Arabia's envoy to Ankara had been summoned to the foreign ministry for a second time on Sunday and had been asked by Turkish diplomats to be "in full coordination" on the matter.
Saudi officials meant to lure Khashoggi back to Saudi Arabia, the Washington Post reported the source as saying, but it was unclear what they meant to do with him - and whether the USA ever warned Khashoggi of the threat he faced.
He used to criticise Saudi Arabia's Crown Prince Mohammed Bin Salman on his adventure against Yemen, arresting women activists and diplomatic ire with Canada.
Turkish authorities believe that a group of 15 Saudi nationals may have been involved in Khashoggi's disappearance.
United States officials have stated that they are following this case closely.More news: Flu shot clinic now open
More news: Don't mention the oil price - US legal threat prompts change at OPEC
More news: Prince Harry and Meghan given royal treatment in Melbourne
US Vice President Mike Pence also commented on the mystery over the missing journalist, saying "the free world deserves answers".
General Assembly President Maria Espinosa Garces is also "very concerned" about Khashoggi's disappearance, spokeswoman Monica Grayley said.
The prince promised social and economic reform, but Khashoggi pointed to the escalating crackdown on dissent voices and the media in Saudi Arabia.
Khashoggi went to the consulate to obtain official documents required for his marriage to Hatice Cengiz.
This reaction, as well as the circulation over Twitter of news stories about the murders of a Kuwaiti businessman and the leader of an Iranian opposition television channel, aim to portray Turkey as a lawless, Islamist-led country - and Khashoggi's disappearance as a effect of that environment, rather than part of a Saudi plot. "There's no difference between the state terror and other terror actions", she added. Saudi Arabia has denied the allegations as "baseless", but offered no evidence to show he ever left the building.
Erdogan has not accused Saudi Arabia of being responsible for Khashoggi's disappearance but has said that if the Saudis have video footage of him leaving the consulate, they should release it. Saudi Arabia is a major investor in Turkey, despite Ankara's support for the Gulf nation of Qatar, which is under a blockade led by Saudi Arabia and three other Arab nations.
Ankara has provided no evidence that Mr Khashoggi was killed in the building. The journalist had been critical of Saudi Crown Prince Muhammad bin Salman.
In a meeting with The Washington Post's publisher, Fred Ryan, the Saudi ambassador to the United States said Sunday night that it was "impossible" that such a crime could be covered up by consulate employees "and we wouldn't know about it".
"Based on their initial findings, the police believe that the journalist was killed by a team especially sent to Istanbul and who left the same day", the government source told AFP on Saturday.
"We demand the worldwide community to pressure Saudi Arabia and Mohammed bin Salman to tell us exactly what happened inside the consulate... and to tell us every detail of who was involved in this crime", Okda told AFP.