Trump's comments followed the publication in pro-government Turkish media of allegations purporting to confirm that Khashoggi was not only murdered by Saudi agents in their consulate in Istanbul, but tortured and dismembered.
"No, not at all, I just want to find out what's happening," Trump told reporters in the White House when asked if his consistently cautious approach to the scandal amounts to a cover-up. "I'm not giving cover at all."
The president said he would get a "full report" from Secretary of State Mike Pompeo -- whom he is to meet at 10:00 am (1400 GMT) on Thursday -- after the diplomat's return from meetings with Saudi and Turkish leaders, allowing him to assess what really happened.
"We will probably know that by the end of the week," Trump said.
The US president has been on the defensive ever since Khashoggi -- a US resident and Washington Post contributor who had been critical of powerful Saudi Crown Prince Mohammed bin Salman -- vanished on October 2 after visiting the Istanbul consulate.
The Post published a column from Khashoggi on Wednesday in which he wrote of the important role a free press could play in the Arab world -- a piece the newspaper admitted appears to be his last.
According to the latest reports, the Saudi journalist was assassinated by a squad that included agents tied to Prince Mohammed, a son of King Salman and a lynchpin in the trend toward ever-tightening relations with Trump's White House.
The controversy has blown a hole in Prince Mohammed's bid to promote himself as the modern face of Saudi Arabia and led to a spate of cancelations by titans of global finance and business at a major Riyadh investment conference scheduled next week.
But Trump has downplayed the possibility of action against Saudi Arabia, which he has repeatedly praised as a major customer for the US weapons industry.
At one point he suggested "rogue killers" could be to blame for Khashoggi's disappearance.
Earlier Wednesday, he told Fox Business that the US relies on the kingdom to fight terrorism.
Pompeo was also tight-lipped after meeting the Saudi leadership in Riyadh, telling journalists he did not want "to talk about any of the facts. They (Saudis) didn't want to either."
Adding to the picture of Saudi influence potentially weighing on American decision-making about Khashoggi, US media reported that $100 million for Washington's stabilization efforts in Syria was deposited by the kingdom as Pompeo arrived in Riyadh.
- Fresh paint in consulate? -
Turkish police and forensic experts on Wednesday searched the residence of the Saudi consul in Istanbul and also searched the country's consulate for a second time.
The consul, Mohammed al-Otaibi, left Istanbul for Riyadh on a scheduled flight Tuesday afternoon, with Ankara insisting he had not been expelled but left of his own choice.
Turkish police had on Monday night carried out an eight-hour search at the consulate, taking away soil and DNA samples. Turkish President Recep Tayyip Erdogan, who also met with Pompeo, said there was evidence that some materials had been freshly painted.
Several US media outlets said Monday that the Saudis are preparing a report that Khashoggi's death resulted from a botched interrogation, but there has yet to be any sign of this being published.
Pro-government Turkish daily Yeni Safak reported it had heard audio recordings of Khashoggi being tortured during an interrogation, having his fingers cut off and then being decapitated.
It said Otaibi can be heard on one tape saying during Khashoggi's torture: "Do this outside. You are going to get me in trouble."
The daily reported that in another tape, an unknown individual tells Otaibi: "If you want to live when you return to Saudi Arabia, be quiet!"
- Senators question Trump-Saudi ties -
The New York Times reported Tuesday that a suspect identified by Turkey was a frequent companion of the prince's. Three other suspects are linked to his security detail and a fifth is a high-level forensic doctor, The Times said.
Adding to embarrassment for the petro-state's royals, International Monetary Fund chief Christine Lagarde became the latest to pull out of Prince Mohammed's much-trumpeted investment conference next week. An IMF spokesman said she had postponed her planned trip to the Middle East with a stop in Saudi Arabia.
US Treasury Secretary Steven Mnuchin said he will decide Thursday whether or not he will attend the conference.
There was also new political pressure on Trump with nine senators from the opposition Democrats writing to express "significant concerns about conflicts of interest" between Trump and Saudi Arabia concerning deals done through his real estate empire.
The letter cited decades of business deals and asked Trump to provide information regarding recent and future financial ties to Saudi Arabia.
Trump defended himself on Monday, tweeting that "I have no financial interests in Saudi Arabia (or Russia, for that matter). Any suggestion that I have is just more FAKE NEWS (of which there is plenty)!"