The journalist, a critic of the Saudi leadership, was last seen alive entering the consulate on 2 October.
"My assessment from these meetings is that there is serious commitment to determine all the facts and ensure accountability, including accountability for Saudi Arabia's senior leaders or senior officials", he said in a statement.
However, no major decision is made outside of the ultraconservative kingdom's ruling Al Saud family.
"This guy's gotta go", he said on Tuesday morning on Fox News, describing Prince Mohammed as a "wrecking ball".
According to Yeni Safak, Saudi Arabia's consul to Istanbul Mohammed al-Otaibi, can be heard on one recording saying during Khashoggi's torture: "Do this outside.
They also indicated they would get this done quickly", Pompeo said before arriving, adding the Saudis vowed that no one would have immunity.
The Saudi Arabian government reportedly sent $100 million to the Trump administration toward its operations in Syria Tuesday - the same day that Secretary of State Mike Pompeo landed in Riyadh - timing that some are calling too good to be true. Al-Otaibi left Turkey on Tuesday afternoon, Turkish state media reported.
Anna Sunik, an Associate of the GIGA Middle East Institute in Hamburg, told DW that no matter how the Khashoggi case turns out, the incident looks "very bad" for the crown prince.
"Here we go again with you're guilty until proven innocent", Trump told the AP in an interview.
Members of US Congress, including South Carolina Republican Senator Lindsey Graham and Democratic Senator Chris Murphy, have called for a change to Washington-Riyadh relations if the Saudi government is indeed responsible for Khashoggi's death.
"I think we have to find out what happened first", Trump said.
Saudi Arabia has said it would retaliate against any pressure or economic sanctions.
According to Erdogan, Turkish investigators who searched the Saudi Consulate in Istanbul found surfaces there newly painted over. The official spoke on condition of anonymity because the investigation was ongoing.
The Saudi consul left Turkey for the Saudi capital Riyadh Tuesday just hours after Turkish officials said they wanted to search his residence and the consulate's fleet of vehicles, Ankara's Foreign Ministry said. Following Trump's comments about the possibility that "rogue killers" might have been involved, several U.S. media outlets reported that a Saudi probe was expected to paint Khashoggi's death as a result of a botched rendition and interrogation. The incident has resulted in a global outrage, more so in the USA as he lived here as a legal permanent resident and worked for "The Washington Post". Under the Vienna convention, diplomatic missions are considered foreign soil.
Meanwhile, Turkish investigators quoted by two news agencies say Monday's search of the consulate has yielded more evidence that Mr Khashoggi was killed there.
Which begs the wider question: is the West's relationship with Saudi Arabia so important that it outweighs the need to condemn and punish what many believe was a state-sponsored murder of a journalist inside a consulate? Khashoggi was a USA resident who wrote columns for the Washington Post and he was critical of the Saudi government, calling for reforms.