Swansea have agreed a fee of £6.2million with add-ons with the Premier League champions for the winger.
The fee could rise to over £8million and the winger is currently holding talks with City over personal terms.
The 23-year-old is keen on leaving Swansea and has told the club that he will not be signing a new deal.
Swansea have signed Valencia winger Pablo Hernandez as a replacement for Sinclair.
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The reputation of the Premier League means it is the most popular and one of the most watched leagues in the world.
Five English clubs will compete in the last sixteen of the Champions League for the first time ever as Liverpool, Manchester United, Manchester City, Chelsea and Tottenham face some difficult away matches.