The 27-year-old English winger is set to make the move to London for around 50 million pounds.
The Blues have been looking to add quality to their frontline with Sterling to team up with Mason Mount, Kai Havertz and Timo Werner.
Chelsea have already seen Romelu Lukaku depart the club this off-season, re-joining Inter Milan on loan after an unhappy year in London.
Sterling was into the final year of his City contract and had expressed a desire for more match minutes, having fallen out of favour at times.
The England international made 339 appearances and scored 131 goals for City across seven seasons after moving from Liverpool in 2015 but often was used off the bench particularly in the 2020-21 campaign.
City boss Pep Guardiola was willing to allow Sterling to leave given he didnt want to keep players who werent happy at the club.
The English champions have also seen Gabriel Jesus exit this off-season, also moving to London to join Arsenal.
But Guardiola’s side made the big money signing of Erling Haaland from Borussia Dortmund to bolster their attacking ranks.
Meanwhile, Cristiano Ronaldo is reportedly set to miss Manchester United’s pre-season tour of Thailand and Australia for personal reasons.
Ronaldo is understood to have handed in a transfer request from United but the reasons cited for missing the tour are a “family issue”.