The 23-year-old has made it clear he does not wish to sign a new deal at the Liberty Stadium, but Swansea rejected a ?6.2million offer from Premier League champions City on Tuesday.
However, a restructured offer of ?6.2million with add-ons that could see the fee rise to over ?8million has now been accepted, and Sinclair is currently holding talks over personal terms.
With Sinclair's departure now imminent, manager Michael Laudrup is set to bolster his wide options after agreeing a fee of around ?5.5million to sign Valencia's Spain international Pablo Hernandez.
Laudrup's pursuit of 27-year-old Hernandez has been a lengthy one, and the Dane had appeared to be exasperated as Valencia continually upped the asking price for a player he previously worked with at Getafe.
Swansea had seen offers of ?4million and ?5.1million rejected, but a third and final bid has been accepted by the La Liga club.
Hernandez had spoken of his desire to be reunited with Laudrup, and was left out of Valencia's squad for their opening two games of the season, and he will now travel to south Wales to discuss personal terms and undergo a medical prior to Friday night's transfer deadline.
Click here for a Live transfer chat with Sky Sports transfer guru Pete O'Rourke from noon.