Mr Khashoggi had entered the consulate to get documents for his planned marriage.
Trump said Wednesday he found King Salman to be a "good man" with whom he speaks often, but acknowledged the Khashoggi case could prove to be an irritant. "It goes without saying that his family in the Kingdom remain gravely concerned about him, and so are we".
News of Saudi Arabia orchestrating the assassination of Khashoggi has put further pressure on its ally the USA to take action.
But Iran, OPEC's third-largest producer, has repeatedly said that its oil exports can not be reduced to zero because of high demand levels in the market and has blamed Trump for an oil price rally caused by imposing sanctions on Tehran.
"This is a bad situation".
European Union's foreign policy chief Federica Mogherini told reporters in Lisbon that "we are 100 percent behind the American position. We can't let this happen".
Corker spoke with reporters after the letter was released, and he emphasized that senators "specifically said it included the highest members of the regime" and could "absolutely" lead to USA sanctions targeting the Saudi Crown Prince, Mohammad bin Salman Al Saud.
All three officials pressed MBS, as the Crown Prince is known, for more details and urged his government to support a transparent investigation, Sanders added. "We will continue to monitor this situation and provide updates as available", White House spokeswoman Sarah Sanders said in a statement.
US Vice President Mike Pence said on Wednesday the US was ready to help in any way in the investigation.
U.S. president Donald Trump, who took his first overseas trip as USA 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 Mr Khashoggi, "but I will be at some point". The passenger manifest, obtained by the pro-government Daily Sabah paper, also lists a senior intelligence officer and two Saudi air force officers.
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While Riyadh claimed he had left the consulate after his visit, Turkish police said Khashoggi did not emerge from the building.
The Washington Post reported on Tuesday that before Khashoggi disappeared, US intelligence had intercepted communications in which the Saudis had been discussing a plot to abduct him.
Mr Khashoggi had sought to become a U.S. citizen after living in self-imposed exile since past year, fearing repercussions for his criticism of the prince, Ms Cengiz wrote.
Reporters Without Borders (RSF), which ranks the kingdom 169th out of 180 on its World Press Freedom Index, said in a statement that between 25 and 30 professional and non-professional journalists are now detained in Saudi Arabia.
"They've got some explaining to do", he said of Saudi Arabia.
Corker says he was told that closed-circuit TV from the consulate was live only and did not record.
"And so to me it feels very much like some nefarious activity has occurred by them", Corker added, according to The Beast.
"He shared with me that they only livestream their tapes", Corker said, according to CNN correspondent Manu Raju.
Kaye said he did not have "any particular plan" about what the global investigation would look like, but that "multiple models" - such as a United Nations -mandated probe - exist.
The Sabah newspaper, which is close to President Recep Tayyip Erdogan, identified the team members, including several alleged security officials, and published photos of each of them, apparently taken at passport control.
A Turkish official, speaking anonymously and without providing evidence, said the 59-year-old journalist was murdered by a 15-member team sent for the task.
Politicians in the United States, Riyadh's main ally, have warned that any harm done to the Washington Post contributor will jeopardize America's relations with the world's largest oil exporter.
Footage also showed some of the Saudis arriving in Istanbul after the first plane landed before 0030 GMT on October 2 and the men later checking into a hotel near the consulate.