They said he was dragged off minutes after entering the consulate to obtain marriage documents. His fiancee, Hatice Cengiz, waits outside.
Mr Khashoggi appeared regularly on Arab talk shows and moved to Washington previous year.
Riyadh insists that he left the consulate safely.
The Saudi ambassador is summoned to the Turkish foreign ministry.
Mike Pompeo, the USA secretary of state, has landed in Ankara for crisis talks with the Turkish president, Recep Tayyip Erdoğan, over the journalist Jamal Khashoggi as officials leaked gory details of what they claim was his murder in the Saudi consulate.
The rights group Prisoners of Conscience, which is an independent non-governmental organization advocating human rights in Saudi Arabia, announced in a post on its official Twitter page that it did not dismiss the possibility that Khashoggi's sudden disappearance was an attempt to silence the writer.
Riyadh calls the claim "baseless". He had this guy murdered in a consulate in Turkey and to expect me to ignore it.
"The President has asked me to travel to Turkey to meet with senior leaders".
Fifteen days after he entered the consulate in Istanbul and was never seen coming out, the Saudis have yet to give an explanation.
In Turkey earlier Tuesday, a high-level Turkish official told the AP that police investigators searching the Saudi Consulate had found evidence that Khashoggi was killed there.
The US State Department says it had not been tipped off about such an operation.
"If the Embassy of Saudi Arabia claims that Khashoggi disappeared after leaving the embassy building, they must prove it", a message quoted President Recep Tayyip Erdogan as saying earlier.
Mr Harbi and Mr Alzahrani have the same names as two people who have been identified as members of the Saudi Royal Guard, the New York Times said.
Saudi Arabia has repeatedly denied any connection to Khashoggi's murder.
Several prestigious partners cancel their planned attendance at the October 23-25 Future Investment Initiative in Riyadh, a lavish conference dubbed the "Davos in the Desert".
The investigators, who eventually searched the premises for eight hours, took samples with them, including soil from the consulate garden, according to an official at the scene.
"I've been their biggest defender on the floor of the United States Senate", Graham said of the country, which has always been an ally of the United States despite its human rights abuses.
Pompeo, wrapping up a trip to Saudi Arabia and Turkey to discuss the crisis over the missing journalist, made a point to stress areas where the kingdom and America cooperate.
Trump says innocence must be presumed.
But the Crown Prince "pledged that the work of the Saudi public prosecutor will produce a full and complete conclusion with full transparency for the world to see", Pompeo said.
Turkish media on Wednesday expected Pompeo to bring answers with him from Riyadh, his previous stop, where he met with King Salman, Crown Prince Mohammed bin Salman and the foreign minister, Adel al-Jubeir.