Khashoggi, a Saudi writer and critic of the kingdom's leadership, was last seen entering the consulate in Istanbul's Levent district on October 2, when he arrived to retrieve an administrative document.
While Riyadh claimed he had left the building afterwards, Turkish police said Khashoggi did not come out of the consulate.
Turkish officials suspect Khashoggi was killed at the Saudi consulate, allegations rejected by Saudi Arabia, which says he left the premises. These include ending all relationships with Iran and Turkey, ending support for the Muslim Brotherhood, and shutting down al-Jazeera, which has been highly critical of Saudi Arabia - things that are never going to happen. It is the only home we know or want. "We have some concrete information, it won't be an unresolved crime", the Middle East Eye reported.
A statement from Turkish spokesman Hami Aksoy said Saudi authorities were "open to cooperation" and had stated their willingness to cooperate and would allow an examination of the consulate grounds.
Britain urged the Saudi government to explain what happened.
"The United States must now make a concerted effort to determine all the facts about Mr Khashoggi's disappearance", the Post said in an editorial, imploring Washington to "demand answers, loud and clear". "If media reports prove correct we will treat the incident seriously - friendships depend on shared values", he added.
One of the two planes that transported 15 Saudis was searched by Turkish authorities before leaving Turkey for Dubai.
Searches of diplomatic buildings are incredibly rare - under worldwide law, the grounds of an embassy or consulate are considered to belong to the country that is represented there, not the host nation.
DailyMail.com said a friend of the journalist revealed that Mr Khashoggi's encrypted messages had been read after he vanished.
The New York Times reported Tuesday that Jamal Khashoggi, an American resident and a columnist for the Washington Post, was killed by the Saudi Arabian government, according to Turkish officials. The situation reached the top diplomatic levels between Turkey and Saudi Arabia. There were no details about when the search would take place.
The Turkish Foreign Ministry has submitted a request to get the permission to inspect the Saudi consulate in Istanbul as part of the investigations of the disappearance of prominent Saudi journalist Jamal Khashoggi last week after entering the consulate. His disappearance sparked global concern.
In this photo taken on December 15, 2014, general manager of Alarab TV, Jamal Khashoggi, looks on durina a press conference in the Bahraini capital Manama.
"Perhaps I'm simply trying to hide from the thought that I have lost a great man whose love I had earned".
"Khashoggi left Saudi Arabia a year ago as the government began its recent crackdown on dissent, arresting clerics, intellectuals, activists and businessmen".
United Nations experts are concerned that Khashoggi's disappearance and the allegations of Saudi Arabia's involvement are related to his criticism of Saudi policies.
The Saudis have called allegations of any involvement in his disappearance "baseless", but had no immediate comment on Turkey's announcement.
Khashoggi was once a Saudi newspaper editor and is a familiar face on political talk shows on Arab satellite television networks. The 59-year-old journalist has a career spanning almost 30 years and was a respected critic of the Saudi regime. Rand Paul said he'd try to push through a Senate vote stalling US arms sales to Saudi Arabia. Relations were already strained after Turkey sent troops to the Gulf state of Qatar past year in a show of support after its Gulf neighbors, including Saudi Arabia, imposed an embargo on Doha.
Turkey reportedly says it is scouring road cameras for a black van believed to have carried Mr Khashoggi's body from the consulate, while the Washington Post released CCTV footage of the journalist entering the building on Monday.