Çalık, which also has interests in energy and finance, appointed Goldman Sachs in January to manage the sale of a controlling stake in its entire media unit, which includes the Sabah newspaper and is partly owned by Qatar.
But while the ATV-Sabah unit could be worth over $1 billion, ATV and Sabah were also being offered separately, sources close to the process said in February.
"Only Abraaj and TNT (Time Warner) are left in the process. They are only negotiating over the television," one of the sources said on Thursday.
"Abraaj wants to buy a majority stake but they want Calik to still have some shares, while TNT wants the whole in a deal in which the Qataris will keep their stake," he added.
A unit of the Qatar Investment Authority, Lusail International Media Co, owns a 25 percent stake in ATV-Sabah, according to Calik's website.
News Corp, Time Warner, Abraaj and Calik all declined to comment.
The Qataris are not selling their stake, the sources said previously, which means they would have to be involved in any move to separate out the TV and newspaper businesses.
However, the sources said it was possible the sale process could also fail due to the declining value of media assets, the time it would take to earn a return on the investment and concerns about political influence on the media in Turkey.
News Corp already owns Fox TV in Turkey and Murdoch told Prime Minister Tayyip Erdogan at a meeting late in March that News Corp aimed to increase its investment in the country, according to government sources.
Çalık Holding paid $1.1 billion in 2007 for ATV-Sabah, one of Turkey's largest media groups.
Ahmet Çalık, chairman of the holding company, is regarded as close to the ruling AK Party's leadership, and that deal gave the socially conservative, economically liberal government an influential friend in the media.
Erdogan's son-in-law, Berat Albayrak, is the chief executive of Çalık Holding, and his brother Serhat Albayrak is the general manager of the media unit.