The Red Devils and Chelsea have both been tipped as potential destinations for the midfielder, but are yet to enter formal talks with the Hammers after the club slapped a £70million price tag on their star man.
The Blues are said to be especially keen on Rice, who was released by the west London club’s academy as a youngster.
United are aiming to beat Chelsea to his signature during the summer, but are not overly optimistic as they believe that Stamford Bridge is his preferred destination, according to ESPN.
The report claims that Rice is aware of interest from United with Ole Gunnar Solskjaer looking to add further improvements in the middle of the park.
The Norwegian boss is aiming to reinforce three areas of his squad by landing a midfielder, striker and central defender before the start of next season.
If a move for Rice is unsuccessful, the Red Devils could turn their attention elsewhere as they are said to have other irons in the fire.
Their prospects of a big-money move could be dented by Woodward’s aim to spend wisely amid the financial effects of the ongoing Covid-19 pandemic.
The United chief executive has reportedly told Solskjaer that he will continue to exercise caution with club funds for the foreseeable future.
Moreover, it is uncertain if Rice will be allowed to leave West Ham at all in the summer.
The midfielder, who made his England debut in March 2019, is contracted until 2024 with the option of extending his stay for a further year.
United legend Rio Ferdinand previously claimed that Rice would prefer to join Chelsea, suggesting that a summer move to Manchester may be difficult to negotiate.
“I don’t think he goes in this window. He stays there [at West Ham] until the end of the season,” Ferdinand told his Five YouTube channel.
“I don’t think he’s the type of kid who is going to push to leave right now. If it happens, it’ll be between the clubs. He won’t be the person to instigate it.
“I think it’s inevitable that he is going to go. If I’m being realistic it’ll be the summer, and it would be Chelsea if he gets his wish.”
Rice has played every minute in the Premier League for West Ham since the beginning of the campaign.
Boss David Moyes revealed in February last year that he is unsure where the midfielder’s future lies, refusing to rule out the possibility of a move in the near future.
“In my mind he is arguably the best holding midfield player in the country and the best thing about Declan is he will get better,” Moyes told reporters.
“Declan is carrying a lot of the team at the moment, and he’s the one who will carry on from Mark Noble when it comes to an end for him.
“Quite often when building a club you have to sell. I hope it’s not the case with Declan, but I said the same with Wayne Rooney and Joleon Lescott [at Everton]. But the team got better because we were able to sign the right players to replace them.
“If that ever came around [Rice leaving], I’d hope we’d be able to do the same here as well.”