The 26-year-old, who has been with the Scottish giants for four years, is hoping to tie up a new deal in the coming weeks.
"I'm happy to say we're getting close to agreeing a contract extension,” Brown told The Daily Record.
"I've said all along I'm settled at Celtic and that I'd love to sign a new deal. Now we're close and I'm delighted about that. I hope the fans see it as good news too.”
The midfielder has claimed he is focusing on doing his best for The Hoops after they missed out on the SPL title to fierce rivals Rangers.
"It's an important year for me. There's no doubt about it,” he added. “I'm 26 and no longer the young lad aiming for the breakthrough. I'm getting on a bit now.
"Seriously, hopefully it's going to be a good season for everyone at the club. We've got to get that title back."
Celtic kicked off their new campaign with a 2-0 away victory over Hibernian.