The Great Britain winger, 23, joins the Blues from Swansea and could be in the squad to face QPR on Saturday.
The Bath-born forward had a terrific season with the Swans last season and was a major factor in the South Wales side’s promotion to the Premier League the season before.
“I’m just glad everything is settled and that I’m now a City player,” said Sinclair.
“There were times when I thought it might not happen so I’m relieved that I’m here and I can’t wait to get started.
“Playing alongside some of the best players in the world is exciting for me and when you see two Champions League fixtures against Real Madrid on the horizon, it brings it all home.”
Sinclair began his career at Bristol Rovers and made his debut aged 15 before moving to Chelsea where he spent five years.
He made a succession of loan moves during his time at Stamford Bridge, playing for Plymouth, QPR, Charlton, Crystal Palace, Birmingham and Wigan before Swansea paid £500,000 for his services in 2010.
Sinclair was an integral part of the Great Britain Olympic squad during London 2012, playing four times and scoring once.
Watch our exclusive interview with Scott.
