The 23-year-old recently told Swans chairman Huw Jenkins and manager Michael Laudrup that he would not sign a new deal at the Liberty Stadium, and with less than a year to run on his current deal he was widely expected to leave before the transfer window closes on Friday.
City were understood to have made an initial bid of around ?5million, but Swansea were keen for a higher offer as Chelsea will get a percentage of any fee due to a sell-on clause inserted when Sinclair moved to south Wales in 2010.
City have now returned with an improved offer, but it is understood it has been turned down by the Swans - who are determined to have their valuation of the player met.
Laudrup and Jenkins have made it clear they are prepared to keep Sinclair in south Wales if they do not believe a deal is in Swansea's best interests.
Sinclair was not included in the squad against West Ham on Saturday, or against Barnsley in the Capital One Cup tonight, but Laudrup has insisted he will restore the wide man to his squad if he remains at the club.
The rejection will only increase City boss Roberto Mancini's frustration as he has voiced his desire to bring in several new faces to strengthen his title-winning squad.
But City now face a race against time to bring in the Team GB star before close of business at the end of the week.
Laudrup remains hopeful of bringing in further additions to his own squad, with deals for Valencia's Pablo Hernandez and Middlesbrough's Marvin Emnes high on the agenda.
Source: PA
Source: PA