And while the Hammers insist Rice is not for sale, the Blues will make a formal bid in excess of £40million before the October 5 deadline.
West Ham co-owner David Sullivan admitted this week he is “getting more depressed" over a lack of signings.
And he conceded that was due to “limited funds".
Hammers boss David Moyes has already been forced to sell winger Grady Diangana to West Brom in a £18m deal that sparked a fans' backlash and a public social media blast from skipper Mark Noble.
But despite spending more than £200m already this summer, Chelsea are confident West Ham can be enticed into opening discussions over Rice, 21.
He remains a close pal of England team-mate Mason Mount after they grew up together at the Blues academy before Rice was released at the age of 14.