President Recep Tayyip Erdoğan on Tuesday said there was strong evidence that Khashoggi's murder was planned.
Visa records are confidential and Pompeo was not more specific about who the revocations would affect, but the State Department later said 21 "Saudi suspects" would have visas revoked or would be declared ineligible to enter the US.
"These penalties will not be the last word on this matter from the United States", Pompeo said during a briefing at the State Department.
Secretary of State Mike Pompeo said the move to revoke visas was just a first step.
"It was a total fiasco", Mr Trump said.
"We have identified at least some of the individuals responsible, including those in the intelligence services, the Royal Court, the Foreign Ministry, and other Saudi ministries who we suspect to have a hand in the killing", Pompeo told the media on Tuesday.
"They had a very bad original concept". US President Donald Trump on Tuesday ridiculed Riyadh's response as "one of the worst cover-ups" in history.
The murder of the Washington Post contributor has severely dented the global reputation of Crown Prince Mohammed bin Salman. "The American people want answers and we will demand that those answers are forthcoming".
Saudi Arabia said it arrested suspects and that several top intelligence officials were fired over the killing, but critics alleged that the punishment was created to absolve Prince Mohammed of any responsibility. Secretary of State Mike Pompeo said the State Department would revoke visas for 21 Saudis the US has identified as being part of the operation that saw Khashoggi killed at the Saudi consulate in Istanbul.
Providing several new details, Erdogan described an operation in which Saudi agents removed the hard disk on a consulate camera and one team visited wooded areas in and around Istanbul "for reconnaissance" before the killing.
Sam Husseini, a journalist who once asked a top Saudi official to defend the legitimacy of his regime, joins for a roundtable on the history of US support for Saudi Arabia, the intentional amnesia of the politicians demanding action, and the slaughter in Yemen.
Turkish authorities claim to have an audio recording that confirms the father-of-four was tortured, killed and had his fingers cut off.