The 26-year old midfielder, who has been the subject of interest from the likes of Bolton and Newcastle United this summer, is poised to sign a long-term contract to remain with the Glasgow giants for the best years of his career.
Brown, who has 12 months remaining on his current deal, confirmed that talks were at an advanced stage and he would soon be committing his future to the Hoops.
"I'm happy to say we're getting very close to agreeing a contract extension,” he said.
"I've said all along I'm very settled at Celtic and that I would love to sign a new deal.
"Now, as I say, we're very close and I'm delighted about that."