Pompeo said he believes those combined efforts will produce "a complete picture" of what happened to Khashoggi, a Saudi journalist living in the USA who had been critical of the kingdom.
Evangelical Christian leader Pat Robertson called this week for Americans to "cool the rhetoric" on Saudi Arabia over the apparent killing of USA resident Jamal Khashoggi because the Middle Eastern nation is an ally and has a lucrative arms deal with the United States.
In a note affixed to the top of the column, Post Global Opinions editor Karen Attiah said she received the essay from Khashoggi's translator and assistant a day after he was reported missing.
"As far as I know, the journalist that disappeared lived in the US".
The Saudi consul general reportedly is heard on tape saying, "Do this outside; you're going to get me in trouble". Khashoggi had left the country past year amid the rise of Prince Mohammed, whom he wrote critically about in the Post.
Turkish crime scene teams were seen leaving the consulate early on Thursday morning.
In the column, Khashoggi, a Saudi citizen who went into self-imposed exile in the US over the rise of the crown prince, also discussed the practice of Middle Eastern governments blocking internet access to control tightly the information their citizens can see.
Citing Turkish law enforcement sources, Al Jazeera, reported that Saudi intelligence elements had "verbally abused and used profanity to address Khashoggi, in addition to torturing him during the first four minutes after he entered the Saudi consulate."
Several US media outlets said Monday that the Saudis are preparing a report that Khashoggi's death resulted from a botched interrogation, but there has yet to be any sign of this being published. "Now we would wait and see", he told reporters.
Despite such heated rhetoric condemning the Saudis, others are suggesting the media seems eager to believe the claims of Turkish officials. "We just need to simply say what happened".
The UK Government spokesman said: "We encourage Turkish-Saudi collaboration and look forward to the Kingdom of Saudi Arabia conducting a thorough, credible, transparent and prompt investigation, as announced".
Pompeo was also tight-lipped after meeting the Saudi leadership in Riyadh, telling journalists he did not want "to talk about any of the facts".
In an interview with Fox News, a prominent Trump ally in the Senate called on Saudi Arabia to reject the crown prince, known as MBS, who rose to power previous year and has aggressively sought to soften the kingdom's image overseas and attract foreign investment.
On Tuesday, a high-level Turkish official told the AP that police found "certain evidence" of Khashoggi's slaying at the consulate, without elaborating.
Dutch finance minister Wopke Hoekstra has cancelled his visit to Saudia Arabia because of the probable murder of journalist Jamal Khashoggi in the Saudi embassy in Istanbul.