A source quoted by The Washington Post said the journalist was killed by a 15-member Saudi team sent "specifically for the murder".
A senior Turkish police source told MEE that Khashoggi had been "brutally tortured, killed and cut into pieces".
The Saudi source said that a security team including Saudi investigators had arrived in Istanbul on Saturday to take part in the investigation into Khashoggi's disappearance.
Fears are growing over missing Washington Post writer Jamal Khashoggi, after Turkish officials said they believe he has been murdered.
Saudi officials insisted he left the building, but Turkey's government said all evidence suggested he was still inside. The source noted the Kingdom's appreciation for the Turkish government's acceptance of the request. Ankara says there has been no evidence to corroborate the 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, Khashoggi has lived in self-imposed exile in the United States since previous year to avoid possible arrest.
"A journalist disappearing like this in a secure country like the Republic of Turkey is something that will be followed up with sensitivity", ruling party spokesman Omer Celik said, according to the Anadolu news agency.
If confirmed, Khashoggi's murder would "constitute a horrific, utterly deplorable, and absolutely unacceptable assault on press freedom", the Paris-based Reporters Without Borders said on Twitter.More news: Pence to Step Up U.S. Criticism of Beijing
In his newspaper columns, Khashoggi has been critical of some policies of Saudi Crown Prince Mohammed bin Salman and Riyadh's intervention in the war in Yemen. Erdogan said on Sunday that he will be following the case closely and is hoping for a positive outcome, adding authorities will be monitoring all camera records and airport transits. I trust Turkey authorities will unveil details.
In an interview with Bloomberg, an American news service, he underscored that Riyadh "will pay nothing" to Washington for Saudi Arabia's security, adding that "all the armaments we have from the United States of America are paid for, it's not free armament". 'We have nothing to hide'.
The Committee to Protect Journalists said it was alarmed by reports Khashoggi may have been killed inside the consulate.
"We do not know if he is alive or not, and the statements by Saudi Arabia on the subject are far from satisfactory", Turan Kislakci, a friend of Khashoggi and chief of the Turkish-Arab Media Association, told supporters on Friday.
Khashoggi may have been considered especially risky by the Saudi leadership, analysts said.
Khashoggi had gone to the consulate to receive an official document for their marriage.
The 59-year-old former government adviser visited the consulate on Tuesday to obtain paperwork to marry his Turkish fiancee, Hatice Cengiz, who was reportedly waiting outside the building for him - but he never emerged.