Palmer, who joins on a seven-year contract with the option of an eighth year, becomes the 11th arrival at Chelsea during a summer of upheaval at Stamford Bridge and sees their spending this window exceed £350 million ($443m).
ESPN reported on Wednesday that Chelsea had agreed a fee with City for the 21-year-old forward. Sources told ESPN on Friday the final fee is £42.5m comprising an initial £40m with a further £2.5m in potential add-ons.
Palmer spent his youth career at City and made his first-team debut in September 2020.
The forward made a promising start to the season at City, scoring in both the Community Shield defeat to Arsenal and UEFA Super Cup victory against Sevilla. However, he had made one substitute in City's opening three Premier League games this season.
"Manchester City has been a part of my life for 15 years and it's all I've ever known since the age of six," Palmer wrote in an Instagram post on Friday.
"I have been very lucky to work with some amazing people along the way. To the coaches the staff, the physios and to all the people behind the scenes. I am very grateful and just want to thank you for helping me become who I am today."