The 16-year-old midfielder has emerged as one of Germany’s most exciting young talents after becoming a regular in the club’s starting XI this season.
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Speaking to Bild before Bayern’s DFB-Pokal final victory over Stuttgart, Eberl admitted the Bundesliga champions are closely monitoring the teenager’s progress.
"I think if you see Kenny Eichhorn - and FC Bayern wouldn't deal with that - then we wouldn't be doing our job," he said.
"Now you have to see if that fits in and what the ideas are for him and his management. Whether that happens then is something else entirely."
Eichhorn made 17 appearances in Germany’s second tier despite his age, attracting strong interest from clubs including Manchester City and Borussia Dortmund.