City confirmed the deal in a club release.
Sky Sports News reported earlier in the day that Sterling had begun his mandatory health checks prior to the announcement.
Rodgers confirmed Sterling's Liverpool exit on July 13, per James Pearce of the Liverpool Echo, and John Cross of the Daily Mirror reported City will pay an initial £44 million with an extra £5 million in add-ons. MailOnline Sport also confirmed the news.
Sami Mokbel of the Daily Mail disclosed the potential details of Sterling's contract, speaking on talkSPORT's Weekend Sports Breakfast show:
I wouldn't expect it to be too far north of [£100,000 a week] to be honest. Obviously Raheem's taken a lot of stick in certain circles over the past few weeks, but this is—from what I understand—a footballing decision. He sees his development and his career away from Anfield, for whatever reason, and I think that's...the main factor in his decision rather than the money.
Sterling was left out of Liverpool's travelling party to Malaysia, Thailand and Australia in order to complete the deal, reported by the Press Association (via ESPN FC).
Sterling was also linked with two other Premier League clubs. Tuttomercatoweb (h/t Anthony Chapman of the Daily Express) claimed on June 24 Chelsea manager Jose Mourinho would make a straight £40 million offer for the England international.
Manchester United were also linked with Sterling, per David Anderson of the Daily Mirror, who reported on June 3 a £25 million bid was rejected by Liverpool.
Sterling's Anfield departure seemed inevitable ever since his agent was quoted saying his client would not sign a new deal under any circumstances.
"I don't care about the PR of the club and the club situation. I don't care. He is definitely not signing (a new Liverpool contract). He's not signing for £700, £800, £900 thousand a week. He is not signing," Ward was quoted saying by Benedict Moore-Bridger of the London Evening Standard.
However, Ward later denied making those comments, per Sky News.
According to BBC Sport's Ben Smith, Sterling informed Rodgers several months ago of his wish to leave Anfield, turning down a new £100,000-a-week contract offer.
Sky Sports News provided further context on the story, with Sterling alleged to feel "bullied" to sign a new deal with the Merseyside giants: