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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Summer 2016 is set to be a very exciting time for football fans, as this is when the long awaited Euro 2016 tournament kicks off.
Since Guus Hiddink arrived at the helm, Chelsea have improved considerably after a tricky start to the campaign.