The pay TV unit of Time Warner Inc. already manages NBA.com, NASCAR.com, PGA.com and part of MLB.com.
A person familiar with the matter says the talks are advanced and a deal is close. The person spoke on condition of anonymity Thursday. The person was not authorized to speak publicly as the discussions are not finished.
Bleacher Report's website says it is the fourth-largest sports media site with more than 25 million monthly readers.
The purchase would enable Turner to sell ads to a broader sports-focused audience and could help boost online traffic and network viewers.
Talks were earlier reported by AllThingsD.com.