Khashoggi, a progressive journalist who left Saudi Arabia previous year, is feared dead by Turkish authorities, who claim he never left the consulate after going in last Tuesday on an administrative errand.
Khashoggi has not been seen since October 2 when he went to the Saudi consulate in Istanbul to obtain official documents for his upcoming marriage.
Saudi officials said he left shortly afterwards but his fiancee, who was waiting outside, said he never came out.
On Monday, Turkey sought permission to search Saudi Arabia's consulate in Istanbul after a prominent journalist from the kingdom went missing last week following a visit to the building, Turkish broadcaster NTV reported.
"The worrying trend that we see is a crackdown on critical and independent journalism in Saudi and about Saudi", Robert Mahoney, Deputy Executive Director of CPJ, told CBS News' Pamela Falk.
Saudi officials have said that Khashoggi left the consulate on his own later that day and that they do not know what happened to him after that.
"If this is true - that the Saudis lured a U.S. resident into their consulate and murdered him - it should represent a fundamental break in our relationship with Saudi Arabia", he tweeted Sunday.
Jamal Khashoggi's fiancée implored President Donald Trump and first lady Melania Trump "to help shed light on" his disappearance in an op-ed published by The Washington Post on Tuesday evening. "It's not possible for us to stay silent regarding an incident like this", Erdogan said.
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Saudi Arabian officials have invited Turkish experts and related officials to visit its consulate in Istanbul.
While unnamed Turkish officials quoted in the media have been giving sometimes macabre details of the alleged murder, Erdogan has so far been more circumspect. "We want to find out what happened".
The Connecticut Democrat says that if Saudi Arabia isn't telling the truth about Khashoggi, then "why would we believe them that they are not intentionally hitting civilians inside Yemen?"
However, CCTV footage shows him entering the embassy but not coming out from it, spurring Turkish President Recep Tayyip Erdogan to demand the Saudis present evidence showing Khashoggi leave the consulate.
The interview briefly touched on the Khashoggi case, with the crown prince stating he had no knowledge of the journalist's fate or whereabouts and that he had left the Istanbul consulate.
The whistleblower organization Wikileaks has massively covered the Khashoggi's disappearance story on Twitter posting updates on latest developments on the story.
Spokesman Ibrahim Kalin says the proposal to set up the committee was made by Saudi Arabia.
A long-time critic of the Saudi government, Paul said he hoped Trump would reconsider his friendly policy towards Saudi Arabia "if there's any evidence they killed this journalist". Khashoggi is a critic of the Saudi regime.