Having established himself as the club's key player and with four years remaining on his contract, Rice is said to be valued in the region of £80m.
However, according to The Sun, the Hammers may have to cash in on the academy graduate if they wish to complete deals for several of their own targets.
The report suggests that the club's board may be willing to entertain a bid of £70m, providing that a proposal is structured in their favour.
Chelsea are known to be monitoring the England international's situation at the London Stadium as they bid to strengthen their options in defensive areas.
Nevertheless, it is previously been claimed that the Blues may only be prepared to go as high as £60m for the 21-year-old's signature.