However, Blues boss Jose Mourinho is not concerned by the rumours and has insisted he wants to keep his Premier League title-winning side together for next season.
The Portuguese tactician said: "I don't see our team losing players. This season is about improving the squad."
The 19-year-old has just one year left on his deal at Stamford Bridge and is believed to be keen on leaving in search of first-team football.
Liverpool are also said to be interested in the versatile defender, but The Sun on Sunday claim Monchengladbach are hoping to get there first with a £3million bid.
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Eden Hazard hopes Christian Benteke joins Chelsea in the summer
The 24-year-old playmaker said said: "If Christian signs with us, I would be the happiest man in the world.
"I have a great relationship with him and would like nothing better than to play with him. He is at an age (24) where he has to make a good career choice."
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