Kushner has fostered a close relationship with Prince Mohammed, the kingdom's de facto leader who has enacted new reforms over the past year even as he's consolidated power and worked to stamp out rivals.
"I don't respond to the critics", Kushner said.
The claim comes a day before Prince Mohammed's high-profile investment summit begins in Riyadh.
For Saudi Arabia's allies - particularly in the West - the question will be whether they believe that the prince, who has painted himself as a reformer, has any culpability.
"We don't blame anyone in advance, but we do not consent to this being covered up", said the ruling-party spokesman, Omer Celik, according to the semiofficial Anadolu news agency.
"Why did these 15 people come here?"
"We can not close this file until the crown prince is brought to account and removed from his post".
In an interview with Fox News on Sunday, he said that Custodian of the Two Holy Mosques King Salman bin Abdulaziz was determined to hold the perpetrators accountable for the murder. Saudi Arabia said he was killed in a fistfight, but Turkish officials said the 59-year-old Washington Post columnist was attacked and killed by a 15-man Saudi team. They are said to have threatened him with being drugged and kidnapped and then killed him in a chokehold when he resisted. "The more one thinks about it, the more it feels like our intelligence is being mocked".
Saudi Arabia so far has not acknowledged or explained Mutreb's presence in Istanbul or the presence of a forensics and autopsy expert at the consulate before Khashoggi arrived. After weeks of denying knowledge of his fate, Saudi officials said the journalist was killed in a "fist fight".More news: Death toll rises to 20 in Jordan school trip flooding tragedy
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Kushner on Monday fended off criticism that the Trump administration was giving Saudi Arabia cover.
In the days after Khashoggi vanished, Saudi officials initially claimed that he had left the consulate, implying premeditation on the part of the Saudi team. Saudi Arabia's preliminary investigation "demonstrates the Kingdom's keenness and commitment to finding the truth of this incident and taking the required legal action against those involved", Egypt's Foreign Ministry said in a statement.
By nightfall, Turkish police began searching an underground vehicle parking garage in Istanbul's Sultangazi district.
Five Turkish employees of the consulate also gave testimony to prosecutors Monday, Turkish media reported.
A senior Turkish official told CNN the surveillance footage of Madani in Khashoggi's clothing adds evidence that the journalist had been executed.
A Turkish official said on Monday that the country will go until the end to enlighten the "nefarious murder" of Khashoggi.
Foreign minister Adel al-Jubeir said that while he did not know exactly how Mr Khashoggi had died, his killing was a "rogue operation" in which Saudi individuals had exceeded their authority.
"We want justice for Jamal", Oke said.