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.
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.
Should Hernandez complete his move he will be Laudrup's eighth summer acquisition, but the former Barcelona and Real Madrid playmaker is still hoping to make further additions.
Swansea are still trying to agree a deal for Middlesbrough's Marvin Emnes, who spent a loan spell at the Liberty Stadium two seasons ago, and are interested in Crewe's Ashley Westwood.
Sinclair, meanwhile, will follow Liverpool signing Joe Allen through the exit door having played an integral part in securing promotion to the Premier League.
Sinclair scored 27 goals after joining from Chelsea in the summer of 2010 and netted a memorable hat-trick in the play-off final win over Reading at Wembley.
He went on to add eight Premier League goals last term as the Swans finished 11th in the top flight.
He will become City boss Roberto Mancini's second summer signing following the capture of Jack Rodwell, and will replace Adam Johnson after the England international's switch to Sunderland.