Khashoggi, 59, who was once close to the Saudi royal family and has served as an adviser for senior Saudi officials, left the country past year to live in the United States in self-imposed exile, saying he feared retribution for his criticism of Saudi policy in the Yemen war and its crackdown on dissent.
The ambassador said the Istanbul consulate is fully cooperating with local authorities and that a Saudi team was sent to Istanbul to cooperate with their Turkish counterparts. "But it seems clear that Saudi Crown Prince Mohammad bin Salman feels emboldened by the Trump administration's unquestioning support".
CCTV footage showed Khashoggi entering the consulate - but there is no public evidence that he either left or was kidnapped or killed.
A surveillance image has surfaced, showing Khashoggi entering the consulate October 2.
The Saudi crown prince added that Khashoggi had left the building not long after he entered.
They had arrived from Riyadh, the Saudi capital, early that morning and checked in at two worldwide hotels in Istanbul before driving to the consulate in the leafy Levent neighborhood, said two people with knowledge of the investigation.
"It is like 'Pulp Fiction, '" the official said.
There was no immediate comment on the report from the Saudi authorities.
Saudi Consul-General Mohammad al-Otaibi said Saudi officials were searching for the journalist and gave a tour of the consulate to Reuters reporters to prove Khashoggi was not there.
"If a person walks into premises and if he isn't stepping out, then who has to prove it?"
"I don't think I'll be able to go home", he told the BBC, saying that in Saudi Arabia "the people who are arrested are not even dissidents" and saying he wished he had a platform at home to write and speak freely at this time of "great transformation" in his country. "Right now, nobody knows anything about it", he said.
On Monday, Turkish President Recep Tayyip Erdogan said the Saudi consulate "must and will" prove where Mr Khashoggi is, saying the Saudis "cannot save themselves" by claiming the outspoken journalist left the building.
Accusations the Saudi leadership directly ordered the alleged assassination of Khashoggi will put further pressure on the United States and other allies to demand a transparent investigation, with possible serious repercussions to bilateral relations if it doesn't come to fruition. He had been living since past year in the United States, in self-imposed exile, in part due to the rise of Saudi Crown Prince Mohammed bin Salman, who has shown little tolerance for criticism.
The newspaper also claimed that Turkish employees at the residence were "hastily" told to take a holiday on the day Khashoggi went missing.
"There is a video of the moment of him being killed", Kemal Ozturk, a columnist in a pro-government newspaper and the former head of a semiofficial news agency, said in an interview on a pro-government television network, citing unnamed security officials.
On Sunday, Turkish President Erdogan said that it was very saddening that Khashoggi disappeared in Turkey.