The 19-year-old has many admirers ranging from Liverpool to Real Madrid, who could provide sizeable competition for Chelsea, but Roman Abramovich has been involved in talks with Maurizio Sarri and external clubs as he arrived in Nice to watch their pre-season match against Inter.
Abramovich has his boat and property in Antibes, close to Nice's Allianz Riviera stadium, where Chelsea won on penalties after a 1-1 draw in the International Champions Cup.
Juventus's Fabio Paratici was in Nice to meet Abramovich and his director Marina Granovskaia, who followed up talks as Gonzalo Higuain, Daniele Rugani and Mattia Caldara remain interests for the club .
Goal understands Pulisic is the newest name to join Chelsea's hit list with the feeling that Dortmund are ready to sell the star with only two years left on his deal.
Sarri may need to move on some of his stars to make room but Willian, having been close to a £60 million move to Barcelona, has now seen that option collapse after they signed Bordeaux's Malcom for £36.5m .
Manchester United may follow up their long-term interest in Willian, although they have yet to make an approach.
Meanwhile, Alvaro Morata has been of interest to AC Milan who are struggling to meet the asking price.
Morata performed well in Chelsea's latest pre-season game despite not scoring and Sarri chose to praise his striker when asked about Higuain by an Italian member of the press.
“I’m still waiting for the best Morata. In the first match Morata did not play very well, in Australia," Sarri said in Nice. "But tonight, especially in the first 30 minutes, he improved a lot. I am satisfied and I am very confident in him for the future.”