Turkish investigators believe a 15-member team "came from Saudi Arabia".
The sources did not say how they believed the killing was carried out.
The Turkish official news agency reported Saturday that the public prosecutor's office has started an investigation into his disappearance and the rumors that Khashoggi was detained had "deepened".
A Turkish official told the AP late Saturday that an "initial assessment" by police concluded Khashoggi had been killed at the consulate. "We believe that the murder was premeditated and the body was subsequently moved out of the consulate", one of the two Turkish officials told Reuters.
"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, of PEN America.
"Saudi Arabia's appreciation for the brotherly Turkish government's acceptance of the request".
The Post said that neither source provided evidence to support this claim.
A former government adviser who has been critical of some policies of Saudi Crown Prince Mohammed bin Salman and Riyadh's intervention in the war in Yemen, Mr Khashoggi has lived in self-imposed exile in the United States since a year ago to avoid possible arrest.
On Saturday, Yasin Aktay, Erdogan's AK Party adviser and a friend of Khashoggi, told Qatari broadcaster Al Jazeera: "We demand a convincing clarification from Saudi Arabia, and what the crown prince offered is not convincing".
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The Prime Minister said he wanted to see stability and prosperity in the region continue. "That's what we're doing". The move prompted a furious response from Beijing, ratcheting up tensions between the two countries.
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The source added that a security team had arrived in Istanbul to investigate Mr Khashoggi's disappearance.
Amnesty International called on Saudi authorities to "immediately disclose the evidence supporting their claim that Jamal Khashoggi left the consulate last Tuesday, otherwise their claims are utterly baseless and only exacerbate suspicions that they are indeed detaining him in what would amount to an enforced disappearance".
"Most of their cases have nothing to do with freedom of speech and a lot of them will return to their homes when the process is finished", he said.
"We are ready to welcome the Turkish government to go and search our premises", he said, which is Saudi sovereign territory. More details from the Turkish authorities are expected on Sunday.
He said that Saudi officials say Khashoggi left the consulate, "but there is no record in the video footage".
The source further said that the interrogators later dismembered his body and the Saudi nationals took interrogation video with themselves to brief Saudi Crown Prince Mohammed bin Salman on his death, the source said. "We have nothing to hide".
Authorities came to see Khashoggi as too progressive and he was forced to resign as editor-in-chief of the Saudi daily Al-Watan in 2003.
'Jamal was - or, as we hope, is - a committed, courageous journalist.
Never one to mince his words, Khashoggi described a new Saudi era of "fear, intimidation, arrests and public shaming" in an article published in the Post previous year.