A team of Turkish and Saudi investigators visited the consulate on Monday, but no official results of the visit have been made public.
Before leaving Riyadh, Pompeo told reporters that Saudi officials pledged they would hold any wrongdoers accountable, no matter how high their positions.
In Saudi Arabia, Pompeo met with Saudi King Salman, the crown prince and the foreign minister. "And I also think it sends... a bad message to the Crown Prince of Saudi Arabia that you can get away with this sort of thing".
"This guy has got to go", Senator Lindsey Graham told Fox.
"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 landed on Tuesday morning in Riyadh and was to immediately meet the king over the crisis surrounding Khashoggi, who disappeared two weeks ago on a visit to the Saudi Consulate in Istanbul. They provided no further details. The kingdom has previously denied allegations that Khashoggi was killed. Maybe - I - I don't want to get into his mind, but it sounded to me like maybe these could have been rogue killers. "We don't make deal - any deals for hostages or prisoners", he said.
"Absolutely extraordinary they were able to enlist the President of the United States as their PR agent to float it".
On Tuesday, Turkish media reported that Saudi Arabia's consul to Istanbul, Mohammed al-Otaibi, had returned to Riyadh as Turkish authorities prepare to search his residence.
Earlier, the Turkish police entered the building of the Consulate General of Saudi Arabia in the framework of the investigation into the disappearance of Saudi reporter Khashoggi. The AP quoted an unnamed high-level Turkish official as saying that evidence was found there of Khashoggi's killing, without elaborating, while Reuters said investigators found "strong evidence" but no conclusive proof of Khashoggi's death. Saudi officials have denied any involvement but have not provided an explanation of what might have happened. Leaked surveillance video shows that diplomatic cars traveled to the consul general's home shortly after Khashoggi went into the consulate.
Turkish forensics arrive to the Saudi Arabia's consulate in Istanbul on October 15, 2018 to search the premises in the investigation over missing Saudi journalist Jamal Khashoggi.
The Saudis earlier insisted, without evidence, that Khashoggi left freely while later reportedly hinted that rogue elements could be responsible - a theory that Trump has mused about aloud.
MBS, who has enjoyed a close relationship with the Trump administration, has painted himself as the face of a new, vibrant Saudi Arabia, diversifying its economy away from reliance on oil and making some social changes.