Khashoggi, 59, had been living in the USA for a year in self-imposed exile and writing columns for the opinion section of the Post.
Crime-scene technicians wore white coveralls, gloves and shoe covers entering the residence.
Another Turkish newspaper, Sabah, has published the names and CCTV pictures of the 15 men who allegedly made a one-day trip to Istanbul on the day Mr Khashoggi disappeared.
Riyadh calls the claim "baseless". "He had [Khashoggi] murdered in a consulate in Turkey, and to expect me to ignore it-I feel used and abused".
Pompeo next travels to Turkey to meet with officials there.
The Russian envoy to Indonesia has said the United States' response to the disappearance of journalist Jamal Khashoggi reflects the double standard the country has always had toward other countries. The secretary of state noted that the Saudi leaders, while denying knowledge of anything that occurred inside the consulate, had committed to accountability "including for Saudi Arabia's senior leaders or senior officials".
Local media report on the possibility that Khashoggi was kidnapped and taken to Saudi Arabia.
Despite concerns about Saudi human rights, Trump still says he is unwilling to pull out of arms sales accords with Riyadh.
CCTV footage released by Turkish TV shows a van entering the consulate on October 2, before going to the nearby consul's residence. "We just need to make sure that we are mindful of that when we approach decisions that the United States government will take when we learn all the facts associated with whatever may have taken place".
Also on Tuesday, U.S. President Donald Trump denied having any financial interests with the Saudi side, to which the U.S. media had attributed the president's ambiguity towards the case.
Mr Erdogan said: "The investigation is looking into many things such as toxic materials and those materials being removed by painting them over". But Saudi Arabia strongly rejects he was killed inside the mission.
Several prestigious partners cancel their planned attendance at the October 23-25 Future Investment Initiative in Riyadh, a lavish conference dubbed the "Davos in the Desert".
Turkish police had on Monday searched the consulate building itself for over 8 hours, taking away soil and DNA samples for further examination.
After his meetings with the king and crown prince on Tuesday, Pompeo said Saudi Arabia has committed to conducting a full investigation.
Trump says innocence must be presumed.
Turkish authorities claim they have video and audio recordings indicating Khashoggi was first interrogated in the consulate, then tortured and murdered.
The top U.S. diplomat said that in Saudi Arabia he stressed the "importance of them conducting a complete investigation into the disappearance" and Riyadh had vowed to do this.
The newspaper said three other people have been linked by the Times to the Crown Prince's security detail, and another one has has been identified as a forensic doctor who specializes in autopsies and who held a senior position in the Saudi Interior Ministry.
The top U.S. diplomat will be in Turkey today to meet Foreign Minister Mevlut Cavusoglu.