The call from Mohammed bin Salman took place before Saudi Arabia acknowledged that the writer was murdered and urged Kushner, President's Trump's son-in-law and a White House adviser, and Bolton to maintain the alliance between the USA and Saudi Arabia, the Washington Post reported.
Saudi Arabia has denied the media reports.
Israel and Saudi Arabia do not have diplomatic relations but share intelligence about their mutual enemies, especially Iran.
He recalled that the German journalist was released after a phone call made by Chancellor Angela Merkel and detained United States Pastor Andrew Brunson was freed after a call by U.S. President Donald Trump.
That version has aroused considerable skepticism among Saudi watchers, considering bin Salman's vast control over what happens in the kingdom.
U.S. secretary of state Mike Pompeo has said it may take "a handful more weeks" before Washington has enough evidence to impose sanctions on individuals responsible for the killing of Saudi journalist Jamal Khashoggi.
"Jamal Khashoggi was not a risky person in any way possible".
Also on Thursday, State Department implicitly urged Saudi Arabia to reveal the location of the body of slain dissident journalist Jamal Khashoggi and return it to his family for burial as soon as possible. "He denied such claims repeatedly over the past several years", the family said.More news: Super Smash Bros. Ultimate Announces New "Spirits" Mode
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However the Saudi chief prosecutor, who visited Istanbul this week, said Saudi authorities had made no such statement about local collaborators, according to the Istanbul prosecutor's office.
Pompeo told a local radio station in St Louis that the U.S. had "deep and long-term strategic relationships" with Saudi Arabia and said "we intend to make sure that those relationships remain intact".
Mr Khashoggi was a Saudi reporter in self-exile in Turkey and a prominent critic of the Saudi regime.
Interestingly, they also met the heads of Saudi Arabia's so-called Global Center for Combating Extremist Ideology (Etidal) and the Muslim World League.
On Wednesday, France's Foreign Minister Jean-Yves Le Drian said Khashoggi's death was a "crime" and "odious".
But analysts say the relationship between Saudi Arabia - which has at times tolerated and abetted the loosely defined political movement - and the group is more nuanced.
Turkish officials said chief prosecutor Irfan Fidan failed to get answers about the location of Khashoggi's body and who ordered his killing during three days of a joint Turkish-Saudi investigation in Istanbul.
What has his fiancée said?
"We must all send a clear message that authoritarian regimes can not kill journalists ever again".