Blues boss Thomas Tuchel is a big fan of Sterling and new owner Todd Boehly quickly got to work on a deal for the England international.
Talks over a fee for Sterling, who has one year remaining on his City contract, continue and it is said to be in the region of £45 million.
Romelu Lukaku's loan to Inter Milan means midfielder N'Golo Kante is the highest earner at Stamford Bridge on around £290,000-per-week. Sources have told ESPN Sterling will be earning more than that.
Chelsea are hoping to seal a deal soon and there is hope Sterling can be part of their pre-season tour to the United States when they depart on Saturday.
The 27-year-old held talks with City at the end of the last campaign and agreed a move was best, although Pep Guardiola would have rather kept the winger.
Sterling started 23 times last season but saw his influence in big games decrease.
He did not start either leg of City's Champions League semi-final with Real Madrid and only started six of their final 11 Premier League games.
Guardiola has strengthened his forward line with the capture of Erling Haaland from Borussia Dortmund. Julian Alvarez, who was signed in January and loaned back to River Plate, is expected to play a significant role next season.
Chelsea finished third in the Premier League and lacked a focal point up front, with Lukaku's £97.5 million transfer ending up a disaster.
The Belgium international expressed his concern about Tuchel's style of playing during a controversial interview in December and his relationship with the fans never recovered.
After sealing a return to Inter, who he won Serie A with in 2021, Lukaku claimed to have been in contact with manager Simeone Inzaghi throughout the last year.
He said: "The affection of the fans and my teammates, but also the opportunity to work with the coach.
"I remained in contact with him throughout all of last season."
Chelsea have also seen centre back duo Antonio Rudiger and Andreas Christensen depart as free agents.
They are reportedly working on a deal for Juventus standout Matthijs de Ligt, although Bayern Munich remain interested too.
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