Turkish officials believe Mr Khashoggi was killed by a team of Saudi agents inside the consulate, with his body then removed.
But one of the key figures in the Saudi-U.S. alliance is receiving particular scrutiny for his role, or lack thereof, in the response: Jared Kushner, the president's son-in-law.
Pompeo declined to comment on what the United States believes might have happened to Khashoggi but made clear Washington takes the situation "very seriously". The consulate building was also searched twice.
Khashoggi, a longtime Saudi critic, has not been seen publicly since he walked into the Saudi Consulate in Istanbul, Turkey, on October 2.
Secretary of State Mike Pompeo said during his visit to the Kingdom this week, the Saudi Arabian government promised a full investigation into the disappearance and presumed death of writer Jamal Khashoggi.
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Kushner told his father-in-law to move forward slowly and not be pressured into any quick decisions on the situation with regards to the possible murder of Washington Post journalist Jamal Khashoggi, a report suggests.
A USA government source said that United States intelligence agencies are increasingly convinced of the crown prince's culpability in the operation against Khashoggi, which they believe resulted in his death.
If the Saudi government is found to be culpable in Khashoggi's disappearance, the legislation prohibits all U.S. military aid and sales to Saudi Arabia until Congress passes a resolution approving such sales.
"Secretary Pompeo has neither heard a tape nor has he seen a transcript related to Jamal Khashoggi's disappearance", spokeswoman Heather Nauert said in a statement.
"We can't abandon these customers - we have to deliver on our promise to be a long-term partner". Increasingly upset USA lawmakers are condemning the Saudis and questioning the seriousness with which Trump and his top aides are taking the matter, while Trump has emphasized the billions of dollars in weapons the Saudis purchase from the United States.
The widening of the investigation came only a few hours after Turkish crime scene investigators finished a nine-hour overnight search of both the consul general's residence and a second search of the consulate itself.