The owner, Burak Akbay, and the three others, including the executive in charge of the website, Mediha Olgun, are accused of links to the movement led by political figure Fethullah Gulen blamed for last year's failed coup, AFP quoted Anadolu news agency as saying.
The fiercely anti-government and ultra-secularist daily whose name means "spokesman" is one of the country's bestselling papers. Its slogan is: "If #Sozcu is silent, Turkey will be silent."
Istanbul prosecutors issued the warrants for the four including correspondent Gokmen Ulu and Yonca Kaleli, a finance executive, CNN Turk broadcaster said.
Anadolu said Olgun had been detained and Akbay was out of the country.
However, Sozcu's lawyer Ismail Yilmaz denied arrest warrants had been issued, telling the private Dogan news agency warrants had been issued to seize and search their belongings.
Yilmaz confirmed Olgun was in custody but said that could be in connection with another investigation, Dogan reported.
The four are accused of “committing crimes on behalf of an armed terror organization,” referring to the Gulen movement, CNN Turk reported.
Turkey refers to the movement as the “Fethullah Terrorist Organization" (FETO), but Gulen vehemently denies ordering the coup and the movement denies any terror charges.
CNN Turk said the suspects were wanted in connection with an online article published on the same day as the attempted coup on July 15 and the accusations leveled at them included “facilitating a real attack on the president”.
The article in question revealed details of where President Recep Tayyip Erdogan was on holiday in the upmarket Aegean resort of Marmaris, CNN Turk said, and had images of his hotel.
Sozcu is the second daily to be targeted after another leading opposition newspaper Cumhuriyet saw 20 staff members charged under the state of emergency imposed after the attempted putsch.
The daily is on occasion rabidly anti Erdogan and its angry front pages are regarded with some suspicion by some liberal Turks critical of the president.
Its sometimes lurid approach contrasts with the more moderate tone of Cumhuriyet, one of the country's oldest dailies.
According to the P24 press freedom website, there are 165 journalists behind bars in Turkey, most of whom were detained as part of the emergency imposed after the coup bid.