Chelsea, Manchester United and Tottenham are also keen on the prodigious 18-year-old but Manuel Pellegrini's men are confident of winning a hotly contested battle.
City first registered its interest in January and the subject of Shaw's availability was broached again when Saints boss Mauricio Pochettino attended the Under-21 Premier League clash between the two clubs at the Etihad Stadium earlier this month, with elite development squad coach Patrick Vieira also present.
The Manchester club was encouraged by the nature of the talks and is now increasingly confident of seeing off a host of rival suitors to secure one of the hottest properties in English football.
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Shaw, 18, is regarded as the finest domestic talent of his generation and is battling Ashley Cole for a place in England's World Cup squad, having made his full Three Lions debut against Denmark earlier this month.
Chelsea is a long-term admirer of Shaw - a boyhood Blues fan - and has a geographical advantage over other interested clubs with the youngster's parents residing in Raynes Park, just a short drive from the club's Cobham training base.
However, the Blues' bid to capture the left back is complicated by the fact that he is represented by Stellar Group, the same management firm which represents Cole, whose contract expires this summer.
City believes Cesar Azpilicueta's impressive performances since displacing Cole at left back have made Shaw wary of summer move to Stamford Bridge.
David Moyes is also in the market for a new left back as he continues to make plans for a summer overhaul of his Manchester United squad and has identified Shaw as a top target.
As reported by Goal, United made a £20 million move for Shaw on January and the Premier League champion is expected to step up its bid to sign him in the next window.
Tottenham is also in the mix to sign Shaw and has launched a charm offensive on the player's family and representatives, who enjoy a strong relationship with chairman Daniel Levy.
Both Chelsea and United have made if clear through third parties that they are prepared to pay up to £25 million to land the Southampton left back, though City's intent takes the potential bidding to a new level.
Sources expect Shaw to join the club that Stellar advise him to, with all signs pointing to him leaving St Mary's Stadium this summer.