Palmer agreed a seven-year contract with Chelsea, who have a club option of a further 12 month extension and could pay City an extra £2.5 million in add-on clauses.
City were keen to keep Palmer, but the size of Chelsea's offer eventually persuaded them to cash in on a rising star who was keen for more regular first-team opportunities.
The 21-year-old could make his Chelsea debut in Saturday's Premier League match against Nottingham Forest at Stamford Bridge.
"I’m excited to get started and it feels great to sign," Palmer said.
"I’ve joined Chelsea because the project here sounds good and because of the platform I will have to try to showcase my talents. It is a young and hungry squad and, hopefully, we can do something special here.”
Palmer's move to the Bridge takes Chelsea's spending in the 16 months since Todd Boehly's Clearlake Capital consortium bought the club to over £1billion.
He is the 12th player to be signed by Chelsea since the end of last season, with his arrival coming after Blues boss Mauricio Pochettino stated he was keen to sign another forward.
Meanwhile, Manchester United signed Turkey goalkeeper Altay Bayindir from Fenerbahce on Friday in a deal worth a reported £4.3 million (R102.14 million).
Bayindir agreed a four-year contract with the Premier League club, who have the option of a further 12 month extension.
The 25-year-old will serve as cover for United's first choice keeper Andre Onana following Dean Henderson's move to Crystal Palace this week.
When Bayindir makes his United debut, he will become the first Turk to play for the Old Trafford outfit.
"It is a huge honour to join Manchester United and become the first Turkish player to represent this incredible club," Bayindir said.
Passion for success
"I have a passion for success, and I will give everything to help this special group of players to achieve our ambitions.
"I cannot wait to start working with such an experienced goalkeeping unit. We will support each other and drive high standards every day so that each of us is ready to perform whenever called upon.”
Bayindir has made 166 appearances in his career, with 53 clean-sheets.
Still with United, Joe Hugill seems to be staying put at the Manchester side.
Elsewhere, Tottenham Hotspur had their off rejected by Bournemouth for defender Lloyd Kelly.
Nottingham Forest have bagged the signature of Nuno Tavares on loan from Arsenal.
On Thursday, Brighton signed Ansu Fati. This comes as Brighton fought off interest from giants Chelsea for the Guinea-Bissau-born forward. The 20-year-old was set for a stellar career at Barcelona before a serious knee injury derailed his career path, but could re-ignite his star in the EPL.