The 19-year-old is highly regarded at West Ham and the club are desperate for him to sign a new contract, with his current deal set to expire in the summer of 2020.
The Hammers are concerned that Rice could leave on a free transfer as he has already rejected their improved offer of around £20,000-a-week.
As exclusively revealed by Metro.co.uk last month, Chelsea have already registered their interest in signing Rice and are willing to double West Hams latest contract offer.
Rice would offer a versatile option to Chelsea as hes capable of playing either as a centre-back or in midfield.
But United are also keen on Rice and have been monitoring his progress throughout this season.
However, Saturdays match against Manchester City was the first time United representatives have watched Rice in person against top opposition.
Although West Ham were comfortably swept aside by the Premier League champions, Rice impressed once again as he sat in the holding role in front of the back four.
City boss Pep Guardiola was also full of praise for Rices display against his side.
Declan Rice is an excellent young player and he will be a top English player for many years to come, Guardiola said after the match.
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