Manuel Pellegrini's side are keen to finally strike a deal for the England forward and there is a belief that their new proposal could see a switch completed in the coming days
Manchester City are prepared to make a final take-it-or-leave-it offer of £45 million for Liverpool's wantaway forward, Raheem Sterling.
City have so far seen initial package offers of £30m and £35m dismissed but believe that an offer comprising a larger up-front payment of £40m plus add-ons amounting to around £5m could get the deal done.
Sterling, for his part, has made it clear to the Liverpool hierarchy that he wants to leave - though he has yet to go as far as submitting an official transfer request.
The 20-year-old could yet do so in a bid to force Liverpool's hand, with the player thought to be increasingly desperate to resolve his exit from Anfield.
Liverpool manager Brendan Rodgers would be happy to sanction a sale at £45m, with his eye cast firmly on bringing in Nathaniel Clyne and Christian Benteke.
The manager - whose power base at the club has been reduced as part of a management reshuffle following last season's sixth-placed finish - is concerned that the Sterling saga could scupper his bid to bring the Southampton right-back and Aston Villa forward to Anfield if the situation continues unresolved through the summer.
With James Milner and Danny Ings already signed up and Joe Gomez coming in from Charlton, Rodgers hopes that the signings of Clyne and Benteke will really see his squad start to take shape ahead of the new season.
City are aware of this but refuse themselves to be hamstrung in their efforts to overhaul their squad by getting caught up in a summer-long bidding war.