The move comes in response to the disappearance of Jamal Khashoggi, a dissident columnist who was last seen entering the Saudi consulate in Istanbul two weeks ago.
The comments marked a break from the Trump administration's previous refusal to speculate over the fate of Khashoggi and came as the USA president is under growing pressure to take action on the case.
"He didn't really know, maybe - I don't want to get into his mind but it sounded to me - maybe these could have been rogue killers".More news: Meeting between Xi and Trump would play 'key role' in easing tensions
On Monday, Trump tweeted that he "just spoke to the King of Saudi Arabia who denies any knowledge of whatever may have happened 'to our Saudi Arabian citizen.' He said that they are working closely with Turkey to find answer".
The case has provoked an worldwide outcry against Saudi Arabia, the world's top oil exporter, with more media and business executives pulling out of a planned investment conference there this month.
Months before Saudi Arabia was accused of sending a murder team to torture and assassinate Washington Post journalist Jamal Khashoggi, the United States Senate had an opportunity to withdraw American military support for the kingdom's vicious, years-long assault on Yemen-but 45 Republicans and 10 Democrats joined hands to squander it.
Turkish officials believe Mr Khashoggi was murdered in the consulate by Saudi agents almost two weeks ago but Riyadh has always strongly denied this. "And incidentally in this whole case of Khashoggi, [Turkish President Recep] Erdogan doesn't come into this case with what we used to say 'clean hands.' He has a awful human rights record". Certain areas of the consulate were to remain off-limits, although officials would be able to inspect surveillance cameras within the post, Turkish media reported.More news: Israel upholds ban on US student refused entry for 10 days
Turkish officials have wanted to search the consulate for days.
Saudi flags, photos of the king and crown prince looking stern, and pro-Saudi hashtags were making the rounds on social media Monday with a common theme: Saudi Arabia and its ruling family are a "red line".
Writing for Al Arabiya yesterday, Mr Aldakhil said: "If US sanctions are imposed on Saudi Arabia, we will be facing an economic disaster that would rock the entire world".
Kristian Coates Ulrichsen, a fellow for the Middle East at Rice University's Baker Institute for Public Policy, warned that the reputational cost of doing business in Saudi Arabia could spike in the wake of Khashoggi's disappearance.More news: Trump Says He Will Call Saudi King About Khashoggi, Calls Case "Terrible"