According to The Athletic, the Premier League champions have sent across an offer of £80 million plus add-ons worth £10 million for the England international.
City's move for Rice rivals that of Arsenal, who saw a club-record bid of their own rejected by West Ham last week.
The Gunners also offered a package totalling £90 million but structured it with a guaranteed £75 million and £15 million in potential add-ons.
Rice is contracted to West Ham until 2024 but the Hammers have the option to extend his deal for a further year.
However, chairman David Sullivan said he expects the 24-year-old to leave the club this summer after their Europa Conference League victory.
Rice made his West Ham debut in 2017 and has gone on to register 245 appearances for the club as well as establishing himself as the club captain.
