The American winger, 20, is valued at around £70million and the Evening Standard say they have now firmed up their interest with an offer.
However, they also claim his first choice would be a switch to Liverpool as he is keen to reunite with ex-Dortmund chief Jurgen Klopp.
Both clubs are understood to be monitoring his situation after talks over signing a new contract with the German giants came to a standstill.
While the report does not outline exactly how much Chelsea are offering, it is claimed it falls below the asking price and that Dortmund are holding out for the full £70m.