Riyadh says Turkish experts and related officials can visit the building, Anadolu Agency reports. Separately, Turkish foreign ministry spokesman, Hami Aksoy, says Saudi Arabian officials have notified Ankara they were "open to cooperation".
Saudi Arabian officials invited Turkish experts and related officials to visit its consulate in Istanbul, the Anadolu Agency reported on Tuesday, following the disappearance of Saudi journalist Jamal Khashoggi a week ago.
Turkey's President Recep Tayyip Erdogan on Monday asked Riyadh to prove its claim that Khashoggi had left the Saudi consulate in Istanbul, while Washington urged Saudi Arabia to support an investigation into his disappearance.
"The Kingdom of Saudi Arabia conveys its regards to the brotherly Republic of Turkey," a diplomatic note sent to the Turkish foreign ministry said.
The note said the kingdom invites Turkish experts to visit the consulate "in line with the cooperation principle" with Turkey "and due to the importance the Kingdom of Saudi Arabia lays on finding the truth about Saudi Arabian citizen, son of Hamza, Jamal Ahmad Khashoggi's incident of disappearance."
Turkey also said on Tuesday that it will search the consulate as part of an investigation into the disappearance of the Saudi contributor to The Washington Post.
The announcement came as the Post published a surveillance image of Khashoggi walking into the Saudi Consulate in Istanbul, just before he disappeared. Turkish officials have said they fear the columnist was killed inside the premises.
Saudi Arabia has called the allegations that it killed 59-year-old Khashoggi "baseless" but has offered no evidence over the past seven days to show that he ever left the building.
'Open to cooperation'
Tuesday's statement from the Turkish foreign ministry's spokesman, Hami Aksoy, said Saudi authorities have notified Ankara that they were "open to cooperation" and would allow the consulate building to be searched. The ministry did not say when the premises would be searched.
The surveillance image released by the Post bore a date and time stamp, as well as a Turkish caption bearing Khashoggi's name and that he was arriving to the consulate.
The Post said "a person close to the investigation" shared the image with them, without elaborating.
The door Khashoggi walked in through appeared to be the main entrance of the consulate in Istanbul's 4th Levent neighbourhood, a leafy, upscale district near the city's financial hub that's home to several other consulates.
However, the consulate has other entrances and exits as well, through which Saudi officials insist he left.
It's unclear which camera the footage came from, nor who operated it.
However, a number of closed-circuit surveillance cameras surround the area.
Friends of Khashoggi say Turkish police have taken possession of footage from the neighbourhood as part of their investigation.
No evidence yet
The Saudis have offered no surveillance footage or evidence to corroborate their claims, nor have Turkish authorities offered proof to show why they believe the columnists was killed there.
"If the story that was told about the murder is true, the Turks must have information and videotape and other documents to back it up," Fred Hiatt, the Post's editorial page editor, told The Associated Press.
"If the story the Saudis are telling, that he just walked out ... after half an hour, if that's true, they ought to have facts and documents and evidence and tapes to back that up."
Hiatt added that the "idea of a government luring one of its own citizens onto its own diplomatic property in a foreign country to murder him for the peaceful expression of his views would be unimaginable."
Khashoggi had gone to the consulate in Istanbul for paperwork to marry his Turkish fiancée. He had been living since last year in the United States, in a self-imposed exile, in part due to the rise of Saudi Crown Prince Mohammed bin Salman, the son of King Salman.
'You have to prove it'
As a contributor to the Post, Khashoggi has written extensively about Saudi Arabia, including criticising its war in Yemen, its recent diplomatic spat with Canada and its arrest of women's rights activists after the lifting of a ban on women driving. All those issues have been viewed as being pushed by Prince Mohammed, who similarly has led roundups of activists, businessmen and others in the kingdom.
Turkish President Recep Tayyip Erdogan on Monday urged the Saudis to back up their claim that Khashoggi left the consulate.
"Now when this person enters, whose duty is it to prove that he left or not? It is (the duty) of the consulate officials," Erdogan said during a visit to Hungary. "Don't you have cameras and other things? Why don't you prove it, you have to prove it."
Turkey summoned the Saudi ambassador on Sunday to request the kingdom's "full cooperation" in the investigation, a Foreign Ministry official said. The Turkish private NTV television said Ankara asked for permission for its investigators to search the consulate building, but a Foreign Ministry official would not confirm the report.
The officials spoke on condition of anonymity because they were not authorised to talk to reporters about the subject.
TRT World's Francis Collings reports.
Saudi Arabia facing increased scrutiny
Ties between Ankara and Riyadh are at a low point over Turkey's support for Qatar in its year-long dispute with Saudi Arabia and other Gulf nations. Turkey sent food to Qatar and deployed troops at its military base there.
Saudi Arabia is also annoyed by Ankara's rapprochement with the kingdom's arch-rival, Iran.
Meanwhile, Saudi Arabia faced increased scrutiny over Khashoggi's disappearance from officials in America, the kingdom's longtime ally.
Even President Donald Trump, who took his first overseas trip as US president to the kingdom and whose son-in-law Jared Kushner has close ties to Prince Mohammed, said he had concerns.
"I don't like hearing about it. And hopefully that will sort itself out," Trump said. "Right now, nobody knows anything about it, but there are some pretty bad stories going around. I do not like it."
UN calls for 'impartial' probe
Meanwhile, the UN rights office has urged Turkey and Saudi Arabia to cooperate on the case.
"We call for cooperation between Turkey and Saudi Arabia to conduct a prompt, impartial and independent investigation into the circumstances of Mr Khashoggi's disappearance and to make the findings public," said Ravina Shamdasani, UN Human Rights Office spokesperson.
Shamdasani said, "At the moment there are still of lot of questions about precisely what happened after Mr Khashoggi entered the consulate, so we will wait until matters are clearer before we can comment further."