The 23-year-old, who represented Great Britain at the London 2012 Olympics, could be in line for his debut in the Premier League encounter with QPR on Saturday.
"I'm just glad everything is settled and that I'm now a City player," Sinclair told City's official website.
"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 is Roberto Mancini's third signing in the transfer window after Jack Rodwell joined from Everton earlier this month and back-up goalkeeper Richard Wright arrived on Thursday.
The trio could be joined by Maicon, Javi Garcia and Matija Nastas.