They are accused of links to the movement led by Islamic preacher Fethullah Gulen blamed for last year's failed coup.
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 Islamic preacher Fethullah Gulen blamed for last year's failed coup, Anadolu news agency reported.
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. CNN Turk said the owner was in London.
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.
Despite the Yilmaz denial, Istanbul prosecutor Irfan Fidan confirmed the operation into "Sozcu's executives", saying "there are warrants, there are detentions".
Fidan did not give details and only told Dogan news agency that the "boss and some workers" were being investigated.
The four are accused of "committing crimes on behalf of an armed terror organisation", referring to the Gulen movement, CNN Turk reported.
Turkey refers to the movement as the "Fethullah Terrorist Organisation" (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 levelled 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.
Izmir correspondent Ulu said no one had come to his home and had learnt of the apparent warrants from the media, the broadcaster said.
"We have fought against terrorist organisations, FETO. As journalists, we have written the truth when they have committed large wrongs against this country," Ulu said.
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 following 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.
The warrants for Sozcu come as Turkey commemorates modern Turkey's founder Mustafa Kemal Ataturk on "Ataturk, Youth and Sports Day".
Sozcu writer Ugur Dundar reacted to the news on Twitter, saying: "On a national holiday, there are detentions from Sozcu. I will go to my newspaper Sozcu. If Sozcu is FETO, everyone in Turkey is FETO!"
Sozcu's front page on Friday had a large image of Ataturk with the headline: "The biggest leader in 100 years" accompanied with the usual image of his eyes next to the paper's name.
The daily issued a statement of defiance on its website: "Let no one have any doubt, Sozcu will not be silent. It will continue to be the voice of this country's conscience."
According to the P24 press freedom website, there are 165 journalists behind bars in Turkey, most detained as part of the emergency imposed after the coup bid.
Turkey ranks 155th out of 180 countries in Reporters Without Border's 2017 World Press Freedom Index.