The Plain Dealer, citing a source, reports the Browns will also hire a defensive coordinator to replace Dick Jauron.
Turner was fired as Chargers coach after a 7-9 season, the team's third straight outside of the playoffs. The Chargers are the third team to fire Turner, who has an overall head-coaching record of 114-122-1.
Rob Chudzinski, who spent the past two seasons as Carolina's offensive coordinator, was introduced as the Browns' sixth full-time coach on Friday.
"Having the assistant head coach experience where I spent all the time with Norv and saw all the things he had to do, I've been around the people who've given me the opportunities to see what it takes to be a great head coach and hopefully I can put all those things together and get that done," Chudzinski said at his introductory news conference.
Information from The Associated Press was used in this report.