The kingdom sacked two top aides to Crown Prince Mohammed bin Salman as well three intelligence officials and arrested 18 Saudi suspects.
While insisting the United States did not condone Khashoggi's killing, the top USA diplomat said the United States had "deep and long-term strategic relationships" with Saudi Arabia and "we intend to make sure that those relationships remain intact".
A Saudi national and well-known critic of Saudi rulers, Khashoggi was killed and his body dismembered on 2 October.
Erdogan said there was more to Khashoggi's death than just action by "a group of security officials", he said.
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.
"Their aim was to leave no trace of the body", Yasin Aktay, an adviser to President Recep Tayyip Erdogan, told the Hurriyet newspaper on Friday, citing the latest information available from the Turkish prosecutor's office.
Istanbul's chief prosecutor announced Wednesday that Khashoggi was strangled immediately after he entered and that his body was dismembered and removed from the consulate.
A vehicle with a diplomatic registration plate leaves the site of Saudi Arabia's consulate in Istanbul, Turkey, Oct. 31, 2018. The worldwide community refused to accept the Saudi claim that the incident was not a premeditated murder. The Saudi attorney general then acknowledged that Khashoggi was killed in a premeditated murder.
US President Donald Trump said Saudi authorities staged the "worst cover-up ever" but has also made more conciliatory remarks that highlight Riyadh's role as a US ally against Iran and Islamist militants, as well as a purchaser of US arms.
Rather than weighing Prince Mohammed's chances of retaining power, officials in the White House and around the region have turned instead to the question of what effect the lingering stigma of the killing may have on the grand plans laid out by the 33-year-old prince, who could rule Saudi Arabia for the next half century.
"With this tragedy, the Trump administration has taken a position that is devoid of moral foundation", she wrote.
Trump has called the affair "one of the worst cover-ups in history", but warned halting a Saudi arms deal would harm United States jobs.
Khashoggi disappeared October 2 after visiting the consulate in Istanbul to get documents he needed to marry his Turkish fiancee.
"Today I am inviting the global community to take serious and practical steps to reveal the truth and to prosecute those involved in a court of law".
But Ms Cengiz stressed that she was "not naïve", and aware that the global response to her fiancé's death would depend on each government's economic and strategic relationship with Riyadh.