The same story says that Dortmund are open to selling the 19-year-old this summer because his current contract expires in 2020 and they would want to get the highest transfer fee possible for the U.S. global.
And now there are claims that Pulisic has been attracting interest from Italy this summer, as the cash received from selling Alisson burns a hole in Roma's pocket.
Pulisic is one of the future stars of world football, and would be an excellent signing for either Real or Chelsea.
The Premier League rivals have reportedly been keeping tabs on the United States of America worldwide over the summer.
The US worldwide managed to clock up a total of 42 appearances in all competitions, bagging five goals and seven assists in the process.
The Stamford Bridge club are hoping to strike a deal this week for a player who has long interested both Liverpool and Bayern Munich.
Pulisic, who scored five times and provided seven assists last season, also has a huge admirer in former Dortmund boss Jurgen Klopp - indeed the Reds failed with a £11m bid for the youngster two years ago.
Pulisic faces an important year after enduring a disappointing campaign last season.
However, it looks like Chelsea have stolen a march on the Merseysiders after Abramovich held formal transfer talks with the German giants over a potential deal to sign the talented teenager.