The newspaper claims Manchester United showed interest in Rice last season, and remain keen to sign him as part of Ole Gunnar Solskjaer's summer rebuilding job.
It also states that West Ham are desperate to keep hold of the newly-capped England international, but adds that an offer of around £80 million could tempt the Hammers into business.
Manchester United lost Ander Herrera to Paris Sain-Germain at the end of his contract and Rice could prove an effective replacement for the Spaniard.
It is suggested that City are seeking a long-term replacement for Fernandinho.
And the chance to join the Premier League winners, who unlike Manchester United will play Champions League football again next season, could be hard for Rice to ignore.
Manchester United are also said to have bid for Rice's West Ham teammate, the defender Issa Diop.
For how much should West Ham consider selling Rice, if at all?
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