The Washington Post, where Khashoggi had been a regular contributor over the past year, also demanded answers.
In the Fox interview, the president was asked if the US' relationship with Saudi was in jeopardy.
Bob Corker (R-Tenn.), Bob Menendez (D-N.J.), Lindsey Graham (R-S.C.), and Patrick Leahy (D-Vt.) sent a letter to Trump demanding that the USA impose sanctions under the Global Magnitsky Human Rights Accountability Act on any "foreign person responsible for such a violation related to Mr. Khashoggi".
The Washington Post cited several of Khashoggi's friends as saying that the prominent journalist had over the past months had been asked by senior Saudi officials close to return back home from the United States, offering him "protection" and "even a high-level job". It has offered no evidence to support its claim the writer simply walked out of its consulate and vanished despite his fiancee waiting outside for him.
The Sabah newspaper, which is close to Turkish President Recep Tayyip Erdogan, published images of what it referred to as an "assassination squad" apparently taken at passport control. Both the Times and the Post are reporting that a group of 15 men, including an "autopsy expert, presumably there to help dismember the body", arrived from Riyadh on the morning Khashoggi disappeared and left Istanbul later that afternoon.
The version of events described by Turkish authorities is grisly. A man with a light blazer outside the consulate bows slightly as the writer enters the building.
When asked for video footage on Khashoggi, Saudi officials reportedly told Republican Sen. "We can't let it happen".
Mr Erdogan questioned whether it was "possible for there to be no camera systems" running at the building, the Turkish daily Hurriyet reports.
Guardiola won't hesitate to play Sterling at Anfield
One situation and they are alone in front of your goalie, or alone in the box and they shoot and score an easy goal. I don't think about him differently for this game because he's a former Liverpool player.
But State Department deputy spokesman Robert Palladino told reporters they had no such tip.
What's next: As the two sides preserve the veneer of cooperation, Turkish investigators were widely expected to conduct a search at the consulate today after Saudi Arabia granted its permission. "We want to see what's going on there". "We're also confident that he would be transparent with readers about these efforts as they progressed".
Trump told reporters in the Oval Office on Wednesday that he had talked "more than once" and "at the highest levels" with Saudi officials.
Mr Trump, who took his first overseas trip as United States president to the kingdom and whose son-in-law Jared Kushner has close ties to Prince Mohammed, said he had not yet talked to the Saudis about Khashoggi.
"We're working very closely with Turkey and I think we'll get to the bottom of it", Trump said today.
White House press secretary Sarah Huckabee Sanders said national security adviser John Bolton and presidential senior adviser Jared Kushner spoke on Tuesday to Crown Prince Mohammed about Khashoggi.
"I recognised his photograph immediately", said Ghanem al-Dosari, a Saudi satirist living in exile in London, who knew Mr Mutreb socially. When asked if he would consider blocking USA arms sales to Saudi Arabia, Trump said that would hurt the United States, citing such deals as a big part of a booming US economy.
"I would say they're excellent", said Trump before going on a tangent. It would be a violation of worldwide law to harm, arrest or detain people at a diplomatic mission, he said, and noted that no such thing had ever happened in Turkey's history. "We're probably getting closer than you might think". The administration also relies on Saudi support for several aspects of its Middle East agenda.
The Washington Post reported that us intelligence services intercepted communications of Saudi officials discussing a plan to lure him back to the kingdom.