Press secretary Sarah Huckabee Sanders says Bolton and Kushner spoke to Prince Mohammed on Tuesday.
The newspaper, where Mr Khashoggi wrote columns critical of the Saudi government, said United States intelligence had intercepted communications showing Saudi officials were trying to lure the journalist back to his homeland so they could arrest him.
If either government seems to be in control of the investigation, Kaye said "it is not going to be seen as credible by significant portions of the global community".
Turkey can not remain silent over the disappearance of Saudi journalist Jamal Khashoggi, President Tayyip Erdogan was quoted as saying by Hurriyet newspaper on Thursday, adding Turkey is investigating all aspects of the case.
Khashoggi's fiancee Hatice Cengiz wrote in an opinion piece for The Washington Post that she was "confident in the abilities of Turkish government officials".
She said he did not think the Saudis could force him to stay at the consulate, adding: 'He did not believe that something bad could happen on Turkish soil.
Khashoggi, 59, was last seen entering the Saudi Consulate in Istanbul through the main entrance on October 2. She said the writer first visited the consulate on September 28 "despite being somewhat concerned that he could be in danger".
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After this third and final older-type gyroscope failed, technicians have tried to bring the balky enhanced gyro back online. There are three gyros of the older generation with a history of showing signs of malfunction after 50,000 hours of service.
The Washington Post columnist has not been heard from since October 2, when he visited the Saudi Consulate in Istanbul, amid speculation that he was killed by Saudi authorities.
News channel 24 aired the video Wednesday, suggesting that Khashoggi was inside of the black Mercedes Vito.
The planes belonged to a company based in Saudi Arabia which has links to the state.
The kingdom has called the allegations "baseless", but has not provided any evidence that Khashoggi left the consulate and did not respond to requests for comment Wednesday.
Turkey said on Tuesday it will search the Saudi Consulate in Istanbul as part of an investigation into the disappearance of a missing Saudi contributor to The Washington Post, a week after he vanished during a visit there. Bob Corker of Tennessee that they "only livestream their tapes".
Angry lawmakers likely won't cause the administration to turn away from Saudi Crown Prince Mohammed bin Salman.
The official described a quick and complex operation in which Mr. Khashoggi was killed within two hours of his arrival at the consulate by a team of Saudi agents, who dismembered his body with a bone saw they brought for the goal.
"We called a few other places, these are lower officials, but they said: "We have evidence he was killed in a barbaric way, we will announce it tomorrow or the day after".