Rodgers led the Hoops to a seventh successive Scottish Premiership title, and has the potential to pick up a historic double treble if Celtic can overcome Motherwell in the Scottish Cup on Saturday.
The Northern Irishman was at a photo call to collect his Ladbrokes Premiership manager of the year award, alongside Brown who was also picking up his player of the year accolade.
When Rodgers considered whether Brown was capable of occupying the top job at Celtic one day, he told reporters: ''I don't see any reason why not.
''His legs are still strong, he is still looking after himself and this season he has won everything, every award going, and he has more years left. But there is no doubt, if he wants to go down that route he will need to go away, do his qualifications and study and get his experiences.
''He has a great history here but he knows if he goes into management it is a different sport, you leave your medals and cups at the door as it is a different game. But I am sure that later in life that is something he can head towards.
''The likes of Neil Lennon did it here, Billy McNeill did it here and other players so I am sure he would like to follow in their footsteps.''
Rodgers also commented on Steven Gerrard's appointment as manager at their Glasglow rivals, Rangers. He said: ''I would expect whoever was in charge at Rangers for it to be tougher.
''That's how football works, each year it gets tougher and more difficult and that will be no different, whoever was taking charge whether it was Steven or anyone else.
''The competition was always there and it will be next season and we have to improve again next season which we will do both individually and as a team.''