BILD says Chelsea are following the midfielder's progress and could have an advantage on rivals.
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Karl is represented by former Blues star Michael Ballack, who also played for Bayern during his career.
The 17 year-old is regarded as Germany's next big thing and scored a first goal in the Champions League win against Club Brugge last week. He followed that up with a goal on the weekend against Bundesliga opponents Borussia Monchengladbach.
Along with Chelsea, Manchester City and Arsenal are also watching Karl this season.