Police were searching the homes of Mr Atalay and Mr Oz, the agency added.
Turkish President Recep Tayyip Erdogan speaks to people demonstrating against the failed military coup attempt.
Ankara blames the supporters of US-based cleric Fethullah Gulen for the coup attempt.
A top European Union politician and leading human rights group slammed the government of Turkey on Monday for shuttering more than a dozen private media outlets and arresting 11 journalists at a key opposition newspaper.
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Sabuncu's arrest also came as the government fought an insurgency from the outlawed Kurdistan Workers' Party (PKK).
Istanbul prosecutors, in a statement published by Turkey's state news agency, said that authorities have been pursuing an investigation against the newspaper since August.
A year ago the paper's former editor-in-chief, Can Dundar, was taken into custody over a story published about the Turkish intelligence service allegedly sending weapons to Syrian opposition. "It is unbelievable the National Press Club has found itself speaking out regarding worsening press freedom conditions in Turkey as many times as it has in the last few years, far more than any other nation and certainly far more than should be needful for a North Atlantic Treaty Organisation member and a critical security partner of the United States".
The suspects are accused of committing crimes on the behalf of terrorist groups, namely the PKK and the Fetullah Terrorist Organization, or FETO which, Turkey says, is behind of July 15 coup attempt that martyred 241 people and wounded 2,200 others.
Another veteran journalist, Kadri Gursel, who began writing for Cumhuriyet in May, said on Twitter that his house was being searched and that there was an arrest warrant for him.
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Rights groups say the crackdown, which has led to the dismissal of more than 100,000 people, has gone far beyond the Gulenists and swept up members of the Kurdish minority as well as other government opponents and critics.
Upon Washington's request, Turkey sent another file last August along with evidence to the US that summarized four previously sent files.
A crowd of up to 70 people, including members of the main opposition Republican People's Party (CHP) gathered outside Cumhuriyet's Istanbul offices to protest the detentions.
"The Department of State made this decision based on security information indicating extremist groups are continuing aggressive efforts to attack U.S. citizens in areas of Istanbul where they reside or frequent", the warning said.
Turkey's President Tayyip Erdogan attends a Republic Day ceremony at Anitkabir, the mausoleum of modern Turkey's founder Ataturk, to mark the republic's anniversary as he is flanked by Prime Minister Binali Yildirim (R) in Ankara, Turkey, October 29, 2016. Journalists at the paper were accused of seeking to precipitate the coup through "subliminal messages" in their columns before it happened, Anadolu reported. It was awarded the Freedom of the Press prize by Reporters Without Borders a year ago and received the Right Livelihood Award, known as the alternative Nobel Peace Prize.
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Sibel Gunes, general secretary of the Turkish Journalists' Association, told The Associated Press that some 170 media outlets have been shut down since the attempted coup and 105 journalists have been arrested.