Real Madrid are listening to offers for the Brazil international, who is not in the plans of new coach Xabi Alonso.
The Premier League appears the most likely destination for Rodrygo, where Arsenal, Manchester City and champions Liverpool are all interested.
But Elber wants Bayern to enter the battle, with the German giants searching for a new attacking signing after being snubbed by Florian Wirtz (Liverpool) and Nico Williams (Athletic Bilbao).
The Brazilian said: "Bayern desperately needs another player like that. Not only for his footballing quality, but also for someone who can create a good atmosphere in the locker room, so there's a relaxed atmosphere.
"It would have to be a Brazilian, for example, who already plays in Europe. Rodrygo is a great player."