City have been monitoring developments as Arsenal had two bids rejected earlier this month for the 24-year-old, the latest of which totalled £90 million ($114.9m) comprising £75m ($89.3m) and a further £15m ($19.1m) in add-ons.
Sources have told ESPN that City were waiting for Ilkay Gundogan to decide his future before moving for a new midfielder, and now that he has chosen to turn down a new deal to join Barcelona, Pep Guardiola's side are ready to act.
West Ham value Rice in excess of £100m ($127.6m), and Arsenal are expected to return with a third offer. It remains to be seen how much City are willing to bid but the Premier League champions signalling their interest will likely drive up the fee.
Sources have told ESPN that Arsenal were confident that Rice wanted to move to Emirates Stadium but City will test that confidence, especially given they can tempt West Ham with a player-plus-cash deal.
Phillips is reluctant to leave and would have to be convinced he has no future at the club in order to continue his career elsewhere.
Arsenal have made Rice their top midfield target and believed talks were progressing well with personal terms not thought to be an issue.
However, it is possible City's pedigree -- having just secured a Treble success of Premier League, Champions League and FA Cup -- may tempt Rice into a change of heart.
West Ham enquired about Maguire, 30, in January but were told he wanted to stay and fight for his place at Old Trafford.