The Hoops boss denied reports that Marseille have held talks with the Parkhead club about a potential transfer for the 21-year-old French striker who scored 32 goals last season, his first in Glasgow.
Ahead of their Champions League third qualifying round first-leg clash against Rosenborg at Celtic Park on Wednesday night, for which Dembele is battling to be fit after picking up a hamstring problem, Brown revealed his admiration for the former Fulham player.
However, the Scotland captain believes Rodgers can improve him further.
He said: “For him to score a hat-trick in a Rangers v Celtic match in his first season and going from there to what he is now, he is a top quality player and underneath this manager he is going to get better.
“We have a great manager and, if he stays here, he will definitely improve.
“I am sure the gaffer wants everyone to stay here, so here’s hoping everyone does stay.
“It is a great bunch of lads in the dressing room and we are just going to get stronger the longer we stay together.
“We have two great strikers in Moussa and Leigh (Griffiths) and they are totally different.
“Moussa can hold the ball up and bring other people into play and do great things and you know what Leigh is going to do, he is going to work hard and score 30/40 goals as well.
“So it is two great strikers, great competition.”
Speaking at Tuesday night’s Celtic Park press conference, Rosenborg manager Kare Ingebrigtsen claimed the pressure was on the Scottish champions to get through to the next round - and guarantee European football next season.
He said: “It will be hard for them if they are not assured of European football.
“Everyone in Scotland thinks they should beat us and beat us easily.
“We know that coming here is really difficult. We need to defend well, keep the ball, but also we have to attack.
“An away goal is very important for us. We are here to score goals.”