Turkish authorities were the first to claim that he was lured into the Saudi consulate in Istanbul, Turkey; killed; and then dismembered with a bone saw before his body was smuggled out in bags.
A Turkish diplomatic source said the investigators planned to widen their search on Tuesday to the residence of the Saudi consul.
Pompeo met King Salman and Crown Prince Mohammed bin Salman to discuss the incident, which has sparked global outrage and brought renewed attention on the authoritarian kingdom's human rights record.
Bin Salman was with Secretary of State Mike Pompeo in Saudi Arabia during that call.
Mr Erdogan said: "The investigation is looking into many things such as toxic materials and those materials being removed by painting them over".
"This guy has got to go", said Sen.
"They said it would be a thorough, complete and transparent investigation", he said.
Trump told AP his comment about "rogue killers" was informed by his "feeling" of the conversation with King Salman.
One of the people suspected of being involved in the disappearance of journalist Jamal Khashoggi has been identified as a frequent travel companion of Saudi Crown Prince Mohammed Bin Salman, according to sources cited by The New York Times on Tuesday.
He maintained in a Tuesday interview with The Associated Press that accusing Saudi officials of killing the journalist was a case of being "guilty until proven innocent".
The top USA diplomat on Wednesday refused to blame Riyadh, saying he did not want "to talk about any of the facts" - a day after President Trump said the kingdom must be presumed to be innocent.
Khashoggi has not been seen since he stepped inside the Saudi consulate in Istanbul on October 2 to sort out paperwork for his upcoming marriage to a Turkish woman.
But some USA lawmakers have all but accepted Turkey's version of the events, that a team of Saudi agents arrived in Istanbul and killed Khashoggi when he went to the consulate to pick up documents he needed to marry his fiancée, Hatice Cengiz, a Turkish national who waited in vain for Khashoggi to emerge from the consulate. He is also reportedly linked to the prince's security detail.
The Saudi Arabian government claimed Mr Khasoggi left the building but his fiancee, who waited outside the Consulate for 10 hours, refutes this.
Mr. Trump also said that his former personal lawyer, Michael Cohen, was "lying" when he said under oath that Mr. Trump had directed him to use campaign funds to pay off two women who alleged that they had an affair with Mr. Trump. Yet he has also repeatedly said that any potential US sanctions should not include killing a multi-billion dollar arms deal with the Saudis.