Saudi Arabia has dismissed the allegation as baseless, while the United Nations has called the conservative kingdom to cooperate with Turkey in conducting an "impartial and independent investigation".
Khashoggi, a Saudi writer and critic of the kingdom's leadership, was last seen entering the consulate in Istanbul's Levent district on October 2, when he arrived to retrieve an administrative document.
A report Monday in the daily newspaper Sabah said investigators were also focused on a convoy of diplomatic vehicles that departed from the consulate on the day Khashoggi vanished.
A Turkish security source told Reuters that a group of 15 Saudi nationals, including some officials, had arrived in Istanbul in two planes and entered the consulate on the same day Khashoggi was there, and later left the country. It is very, very upsetting for us that it happened in our country.
A friend of Khashoggi's said officials told him to "make your funeral preparations".
Karen Attiah, an editor and journalist at the Washington Post and Khashoggi's editor also tweeted on the matter calling USA companies to "rethink" their relations with Saudi Arabia.
Among the assassination team was a forensic expert who brought a bone saw to dismember Khashoggi's body after killing him, the Times reported on Tuesday, citing an unidentified American senior official as saying.
Erdogan pointed out that "security and intelligence officials are investigating Khashoggi's case, and the prosecution is checking up the records of the arrival and departure of Saudi citizens from Istanbul airport".
USA knew of Saudi plan to seize missing journalist, newspaper claims
Sabah newspaper published the names and years of birth of the 15 Saudis it said arrived at Istanbul's Ataturk airport. Twelve of them arrived early on Tuesday, based on photos captured at passport control which were published.
Why it matters: The incident threatens a diplomatic crisis in Turkey's relations with Saudi Arabia.
Jamal Khashoggi went to the consulate last Tuesday to obtain a document certifying he had divorced his ex-wife, so that he could marry his Turkish fiancée, Hatice Cengiz.
Cengiz urged Saudi Arabia, "especially King Salman and Crown Prince Mohammed bin Salman, to show the same level of sensitivity and release CCTV footage from the consulate".
"We call on the government of Saudi Arabia to support a thorough investigation of Mr. Khashoggi's disappearance and to be transparent about the results of that investigation", Pompeo said in a statement.
Trump has instead focused on U.S. and Saudi shared interests in ratcheting up pressure on Iran.
The growing dispute over his fate threatens relations between Saudi Arabia and Turkey and raises new questions about the kingdom and the actions of its assertive Crown Prince Mohammed bin Salman, whom Khashoggi wrote critically about in his columns.
A Sunni power, Saudi Arabia is also annoyed by Ankara's rapprochement with the kingdom's Shiite archrival, Iran.
"At this time, I implore President Trump and first lady Melania Trump to help shed light on Jamal's disappearance", Cengiz wrote. Fearing for his life, Khashoggi has lived in self-imposed exile in the United States for the previous year. "I do not like it", Trump told reporters at the White House.
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That Saudi Arabia would allow foreigners to enter a consulate and search it shows the growing global pressure the kingdom faces over the disappearance of Mr Khashoggi, a critic of Saudi Arabia and contributor to the Washington Post.