With his skipper's contract due to expire in the summer of next year, the Cottagers' manager has admitted the club could be forced to listen to offers if they come in January
The 31-year-old defender is yet to agree terms with the club he joined in 2008, meaning Jol may have to consider any bids that are made for him should they come during January's transfer window.
Amid speculation that Fulham may be forced to sell a number of star players after the current campaign, Jol suggested it could only be the Norway centre-back who departs.
|12/1||Fulham are 12/1 with Bet365 to beat Reading 2-0 on Saturday
"Our culture is that we always like to keep the players we want to keep," the Dutchman told Sky Sports. "I don't know where the story that our star players are going is coming from.
"Brede is the only problem, but we've been talking to him for the last five or six months. But we've made him a good offer so if he wants to stay that would be fantastic.
"So believe me, there isn't a situation where a lot of my best players would go. [Dimitar] Berbatov has two years, [Mladen] Petric one year and an option, so that is fine and [Hugo] Rodallega was a free transfer. [Sascha] Riether has an option."
During pre-season, the revolving door at Fulham was in full motion, with key men such as Clint Dempsey, Mousa Dembele and Danny Murphy all leaving the club.
"We always want to improve and don't tell anybody else, but I believe that's what we did in the transfer window," Jol added.
"With Dembele it was different, he left three days before the transfer window closed after promising he'd stay.
"We spoke to Clint 18 months before his contract expired, but he wanted to go to a Champions League club."