ESPN says United are planning on signing a midfielder next summer, and West Ham's Rice is one of those on the shortlist.
The club were well aware of supporters' desire to see a midfield signing in the recent transfer window, but decided to prioritise other positions.
It is believed that the Hammers would have to lower their current valuation of in the region of £100m for the 22-year-old, who is keen to play Champions League football, if a deal is to be struck with United.
Rice is under contract at the London Stadium until 2024, so there is no immediate need for West Ham to consider cashing in one of their star players.
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