Brown has been linked with Newcastle as a replacement for Kevin Nolan.
The 25-year-old told the Daily Record: "I've heard all the stuff about Newcastle but I haven't paid the slightest attention to it.
"I'm captain of Celtic Football Club, why would I even think about going anywhere else?
"A few weeks ago I had the privilege of lifting my first piece of silverware since being given the armband 18 months ago.
"As the gaffer said himself 'this is not the end, it's only the beginning.'
"I have a hunger to win more trophies as Celtic captain so that I can look back at my time at the club and consider myself to have been a success."