The 22-year-old was heavily linked with a move to Stamford Bridge last summer, but ESPN reports that United have now entered the race.
Ole Gunnar Solksjaer wants to add a defensive midfielder to his squad ahead of the 2021/22 season, along with a forward and centre-back, and the Red Devils have shortlisted Rice.
The England international knows of the interest from Old Trafford, but Chelsea are still considered the favourites to sign him should he leave the London Stadium.
West Ham are determined not to let him go without a fight, and will demand £70 million for a young player who still has another three-and-a-half years left to run on his contract, with the option of a further year.
The impact of the coronavirus pandemic means that finances will be tighter than usual for United, but they do intend to invest with caution in the summer.
Rice’s potential move to Manchester could depend on Paul Pogba, whose future remains up in the air since his agent Mino Raiola publicly said he intended to find a new club for the Frenchman.
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