Trump said Pomepo would be departing Washington "literally within an hour or so" and has license to travel wherever necessary, including Turkey, to investigate what happened. Trump first announced Pompeo's trip in a tweet in which he said Saudi Arabia is "working closely with Turkey to find answer."
Trump also said that with Turkey's release of jailed pastor Andrew Brunson, relations were vastly improved.
"I have a very good feeling toward Turkey. Two days ago, I did not," he said.
His comments came hours after three Turkish officials said Saudi Arabia has agreed to allow Turkish investigators to search the Saudi Consulate in Istanbul, 13 days after Khashoggi, a Saudi journalist, vanished upon entering the mission.
Crime scene investigators were pictured entering the consulate. Turkey had publicly scolded the Saudi Government for refusing repeated requests to search the consulate, where Turkish investigators believe Khashoggi was killed and dismembered by Saudi agents.
Saudi Arabia has vehemently denied any knowledge of Khashoggi's whereabouts. The agreement for a search of the consulate came a day after King Salman called Turkish President Recep Tayyip Erdogan, thanking him for welcoming the kingdom's proposal to set up a "joint working group" to probe Khashoggi's disappearance, a Saudi statement said.
Khashoggi lived in self-imposed exile in the US for the past year and wrote columns in the Washington Post criticising the Saudi leadership. He visited the Saudi Consulate on October 2 to obtain documents related to his upcoming wedding, but he was never seen leaving.
The Saudi Government has faced intense pressure to reveal his fate, as Turkish officials have released details of their investigation, including video that suggests a team of Saudi agents was dispatched to Istanbul to either capture Khashoggi or kill him.
The Turkish Government has also told the Trump Administration that it possesses audio recordings of what occurred inside the consulate that day - evidence that US officials said supports the conclusion that Khashoggi was interrogated, tortured and then murdered.
US officials have also said that the crown prince of Saudi Arabia, Mohammed bin Salman, ordered an operation to lure Khashoggi from his home in Virginia back to Saudi Arabia and then to detain him, according to US intelligence intercepts of Saudi officials discussing the plot.