Pulisic is regarded as one of the hottest prospects currently plying their trade in Europe, with the 19-year-old having already contributed 11 goals in 84 appearances for Dortmund.
However, the youngster's performances are already said to have attracted interest from both Manchester United and Liverpool and according to the Daily Mail, Chelsea and Bayern are ready to get involved in the race for the USA international's signature.
Bayern have made a habit of signing some of the Bundesliga's top stars in recent years, and the German giants would allegedly be prepared to spend big in order to take Pulisic to the Allianz Arena.
Chelsea find themselves in a position where they may receive bids for Eden Hazard during the summer, and a long-term replacement for the Belgian playmaker could be required in the future.
The report suggests that Dortmund may demand the permanent signing of Michy Batshuayi in any deal for Pulisic, with Batshuayi currently on loan at the club from the Blues and having netted five goals from his opening four games in all competitions.
Pulisic - who is likely to command a fee of over £80m - still has two-and-a-half years remaining on his Dortmund contract.