The Blues are believed to be ahead of Manchester United in the race to sign the midfielder, who turned down two contract offers from the Hammers last month.
Rice's future remains uncertain but according to the Daily Telegraph, the 22-year-old is still prepared to give his all for West Ham and will not look to force through a transfer.
The midfielder still has three years remaining on his current contract and the Hammers are under no pressure to sell one of their star players, who they value at around £100m.
In addition, manager David Moyes views the England international as a key asset in his squad, who are looking forward to competing in the Europa League group stages for the first time.
As well as starting in all seven of England's Euro 2020 matches, Rice made 35 club appearances across all competitions last season, captaining the Hammers on 23 occasions in the Premier League.