The 23-year-old has informed Swansea chairman Huw Jenkins and manager Michael Laudrup he has no desire to sign a new deal at the Liberty Stadium, and with his current contract set to expire next summer he was always likely to be sold.
City have long been the favourites to sign the former Chelsea man and Press Association Sport understands they have firmed up their interest by making a bid, believed to be in the region of £5million.
Swansea first-team coach Alan Curtis admitted a move to the Etihad Stadium was looking increasingly likely for Great Britain Olympic forward Sinclair, who scored in the thumping 5-0 win at QPR on Saturday.
Curtis told BBC Sport: "He's not really interested in signing a new contract and I think Man City have made a bid.
"So it's just a matter of time now before it happens and it's good for both parties."
While Sinclair is set to head through the exit door, Laudrup's recruitment drive is continuing.
Celtic's South Korea international Ki Sung-yueng is discussing terms with Swansea after the Scottish champions accepted a bid of just over ?5million, although the likes of QPR are thought to be waiting in the wings should the Welsh club fail to land their man.
Winger Pablo Hernandez is another man Laudrup wants to bring to the Liberty Stadium.
The Spain international was left out of the Valencia squad that drew with Real Madrid on Sunday and Swansea remain hopeful a deal can be done.
Source: PA
Source: PA