The news will come as a blow to Arsenal who had a club-record offer worth up to £90million turned down by the Hammers this week.
Arsenal had pinpointed Rice as their main transfer target this summer but the impending exit of City captain Ilkay Gundogan has put the European champions on red alert.
Despite Arsenal leading the race for Rice's signature, City were have kept tabs on the England international and now look set to launch an official bid, according to The Athletic.
Rice has been the target of speculation over his future after failing to agree a new deal at the London Stadium.
The 24-year-old has just 12 months remaining on his deal however West Ham have the option of triggering a further year.
Read more on Man City
Jamie Carragher calls for Saudi Arabia 'sportswashing' to stop amid Bernardo Silva links
Ilkay Gundogan to join Barcelona after turning down one-year Man City offer
Rice was believed to favour a move to north London but that was ahead of City showing a firm interest and now the midfielder could be enticed by the prospect of playing under Pep Guardiola.
After leading the Hammers to a historic Europa Conference League final success last season, chairman David Sullivan admitted his captain was likely to leave the club this summer.
When asked if the win over Fiorentina was his last in a West Ham shirt, Sullivan exclusively told talkSPORT: "I think it has to be.
"We promised him he could go. He set his heart on going. You can't ask for a man who has committed more to us this season
Most read in Football
Kane told to force move, Gundogan to Barca, Newcastle close in on Tonali, EFL fixtures
EFL fixtures announced with Leeds vs Cardiff on opening weekend
West Ham fans hit with European ban after Europa Conference League final carnage
Ilkay Gundogan to join Barcelona after turning down one-year Man City offer
"In due course, he has to get on and we have to get a replacement - or several replacements.
"It's not something we want to happen. We offered him £200,000-a-week 18 months ago. He turned it down.
"It's cost him £10million to stay at West Ham in that time (in lost wages). And he wants to go.
"You can't keep a player who doesn't want to be there."
More will follow.
For the latest on this and other breaking news stories, listen to talkSPORT