The three posed for photos before their meetings, but said nothing in front of journalists.
Following the meeting with Erdogan, Pompeo also had a separate conversation with Cavusoglu which lasted 35 minutes.
Mr Erdogan has previously suggested parts of the consulate had been repainted since the disappearance of the journalist, who was a critic of the Saudi regime.
Turkey believes Khashoggi was deliberately killed in the Saudi consulate and his body was removed on Oct 2. A Saudi delegation reportedly had entered the consulate one hour before the Turkish police arrived and appeared still to be inside as the search was conducted. Kavanaugh was eventually confirmed after the FBI investigation found no corroborating evidence for the charges.
Khashoggi is feared killed after he entered the consulate and was never seen exiting.
The newspaper said Khashoggi, a Washington Post contributor, was then decapitated.
A spokeswoman for the Saudi Embassy in Washington did not immediately respond to a request for comment about the discussions.
Turkish FM Saudi Arabia must cooperate on Khashoggi, allow access to consulate
It relies on Saudi support for its Middle East effort to counter Iranian influence and fight extremism. The Trump administration, from the president on down, is heavily invested in the Saudi relationship.
After speaking with the king, Trump floated the idea that "rogue killers" may have been responsible for the disappearance.
Regan, understandably, asked Trump whether he still plans to send Treasury Secretary Steven Mnuchin to Saudi Arabia for an upcoming high-profile Saudi investment conference despite the "controversy surrounding missing Washington Post journalist Jamal Khashoggi".
The US's defence of its most important Arab ally may become harder to maintain as further damning details into Khashoggi's alleged murder emerge and its links to the powerful crown prince.
Pompeo said the Saudis are committed to a "thorough, transparent and timely" investigation.
Khashoggi, a Washington Post columnist who vanished after visiting the consulate on October 2 to pick up paperwork ahead of his wedding, had his fingers chopped off before he was beheaded, according to the pro-government Yeni Safak. However, in a contrary scenario, the kingdom, highly criticized due for the actions of Crown Prince Mohammed bin Salman, will remain as a rogue state in the global system.
Republican Senator Lindsey Graham said Tuesday the US should "sanction the hell out of Saudi Arabia" over the incident and said he would never again work with the crown prince, assailing him as "toxic" and calling him a "wrecking ball".
"They made no exceptions to who they would hold accountable".