"What we have right now is Mike Pompeo, who's terrific, who's doing a great job, Secretary of State, is with the King right now and the Crown Prince literally as we speak and we're trying to find out what went on", Trump answered.
Turkish crime scene investigators carried out a eight-hour search of the consulate on Monday night in an investigation to look into what happened to Khashoggi, who was last seen entering the Saudi consulate on October 2.
"I don't want to get into his mind", Trump said of the king, a former business associate of the erstwhile real-estate magnate.
Mr Khashoggi had been unafraid to criticise the government and Crown Prince and had previously been warned to stop writing articles and stop using Twitter.
That attitude does not appear to be shared with Congress, as one prominent Republican senator said he believed that the crown prince, widely known as MBS, had Khashoggi "murdered".
That leaves Trump, who Monday suggested it might have been "rogue killers" behind Khashoggi's murder, out of step with even close allies in Washington like U.S. Sen.
"This guy has got to go".
The pro-government paper Sabah has reported that Khashoggi's Apple Watch recorded his interrogation, torture and killing, but some experts have said it was unlikely the watch could have recorded the events in that manner.
"The one thing we really know as a solid fact is that Khashoggi went into the consulate and he was never seen coming out again", Colville said, adding that it seemed "very probable" that some crime had been committed.
Erdogan on Tuesday said he hopes a reasonable opinion will be reached as soon as possible in the investigation, adding the search in the Saudi consulate in Istanbul will continue with the investigation also looking into toxic materials. I'm going to sanction the hell out of Saudi Arabia.
Consul Otaibi left Istanbul on Tuesday for Riyadh ahead of a planned search of his residence as part of the official Turkish investigation into Khashoggi's disappearance. The Saudis initially denied that this happened, but without any video evidence that he ever left the consulate, they've seemingly painted themselves into a corner.
The crown prince "pledged that the work of the Saudi public prosecutor will produce a full and complete conclusion with full transparency for the world to see", Pompeo said.
Earlier Erdogan said that the Turkish intelligence is investigating the disappearance of the Saudi reporter Khashoggi.
Turkish police on Monday night undertook an eight-hour search at the consulate, taking away soil and DNA samples, although a search expected on Tuesday at the consul's residence has so far not taken place.
The Saudi heir apparent wields considerable power in the kingdom and is being held responsible by many outside for whatever happened to Khashoggi.