After overcoming some minor sticking points in final negotiations the clubs were able to come to an agreement on a Sterling 53 million ($A103.7 million) move - a club-record sale for Wolves.
The 25-year-old Portugal international was keen to move to the Etihad Stadium and did not train with Wolves in order to push the transfer through.
"The opportunity to work under Pep Guardiola, one of the greatest managers ever, and alongside some of the best players in the world was something I couldn't turn down,"Â Nunes told City's official website.
Midfielder Tommy Doyle, 21, has moved the other way, joining the Molineux club on a season-long loan with an option to make the switch permanent.
Doyle's exit from the Etihad follows that of fellow City academy product Cole Palmer, who has joined Chelsea on a seven-year contract.
Chelsea will pay an initial Sterling 40 million for the 21-year-old.
The signing of Palmer takes Chelsea's spending in the 16 months since the club was acquired by Todd Boehly's Clearlake Capital consortium to more than Sterling 1 billion.
Manchester United confirmed the signing of Turkey international goalkeeper Altay Bayindir from Fenerbahce.
The 25-year-old has signed an initial four-year contract at Old Trafford, with the option of a further year.
Bayindir is the second goalkeeper signed by the Red Devils this summer and will provide competition to fellow new arrival Andre Onana.
"It is a huge honour to join Manchester United and become the first Turkish player to represent this incredible club," Bayindir said in a statement announcing his signing.
Nottingham Forest signed Arsenal full-back Nuno Tavares on a season-long loan, while Aston Villa brought in French defender Clement Lenglet on a season-long loan from Barcelona.
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