The deal would give Demiroren, which already owns two pro-government newspapers, control over several prominent media outlets including the respected Hurriyet newspaper and broadcaster CNN Turk.
The business has an operating value of $1.1 billion and an equity value of $890 million after debt, Dogan said in a statement to the Istanbul stock exchange.
Shares in Dogan and Hurriyet both jumped by the session limit of 19.5 percent at the start of trade, after rising a similar amount a day earlier.
Hurriyet subsequently trimmed its gains and was 8.1 percent higher at 1.33 lira at 0721 GMT, while Dogan was up 18.4 percent at 1.03 lira.
Dogan Holding's media arm and its octogenarian founder, Aydin Dogan, have long been seen as part of Turkey's secular establishment. Erdogan has repeatedly accused the company of bias against his Islamist-rooted AK Party, which the company has denied.
Dogan Media was fined $2.5 billion for unpaid taxes in 2009, in what government opponents saw as an attempt to crush media criticism of Erdogan. Following the tax bill, Aydin Dogan was forced to sell the group's Milliyet and Vatan newspapers - to Demiroren.
The papers have since adopted a strong pro-government stance.
Rights groups and some of Turkey's Western allies have sharply criticised the country's record on media freedom and human rights following a widespread crackdown after a failed coup in July 2016.