"I just spoke with the King of Saudi Arabia, who denies any knowledge of what took place with regard to, as he said, his Saudi Arabian citizens," Trump said on the South Lawn of the White House. "I've asked -- and he firmly denied that. We are going to leave nothing uncovered.
"Maybe -- I don't want to get into his mind -- but it sounded to me like maybe these could have been rogue killers. Who knows?
"The King told me that Turkey and Saudi Arabia are working hand-in-hand, very closely getting to the bottom of what happened."
Pompeo could also travel to Turkey as part of the investigation, Trump said.
Turkish officials said Monday they were given permission to search the Saudi consulate in Istanbul where Khashoggi went missing and, according to some investigators, was killed and dismembered.
The search could happen as early as Monday afternoon, 13 days after Khashoggi walked into the consulate to get a document for his upcoming marriage.
"There's a lot at stake," Trump told CBS News' 60 Minutes Sunday. "And maybe especially so because this man was a reporter. There's something, you'll be surprised to hear me say that, there's something really terrible and disgusting that if that were the case.
"We're going to get to the bottom of it and there will be severe punishment."
Trump said he didn't want to hurt potential jobs and was reluctant to jeopardize a $100 billion arms deal with Riyadh, signaling that he might not use sanctions against Saudi Arabia. Some have also expressed concern about the case's impact on world oil markets.
"There are other ways of punishing," Trump said.
Khashoggi has lived in self-imposed exile in the United States for the past year, writing columns in The Washington Post that criticized the Saudi regime. He visited the consulate in Istanbul on Oct. 2 to get documents related to his planned wedding.
Riyadh said through its state-run news agency it would retaliate against any potential sanctions in the Khashoggi case from the United States or other governments.