Manchester United this week expressed an interest in Rice, who made 34 Premier League appearances last season.
The 20-year-old excelled at the base of Manuel Pellegrini’s midfield and earned his first senior England cap in March, catching the eye of Ole Gunnar Solskjaer.
The Norwegian wants to build a crop of young, preferably English, talent this summer and Rice fits the bill.
Sky Sports News understand that West Ham have told United, and Manchester City, they have no intention of selling Rice this summer, though.
Rice calmed rumours yesterday by responding to a West Ham Instagram post about him, writing: “See you all soon hammers.”
Speaking about his future last month, the pivot said: “Everything’s good. I know there’s speculation but during the season I signed a new five-year contract with West Ham, so my focus is fully on playing for West Ham.
“I have a great connection with the fans. The fans love me, I love them, and that’s the way it is at the moment. I’m not thinking about anything else.
“I think when you start out in football for sure you always want to play at the top, you want to play in the Champions League.
“I think if you were in football and didn’t want to do that, there’d be no point in playing.”
If United do change tact in the market it could mean veering towards defensive reinforcements as opposed to midfield.
Crystal Palace right-back Aaron Wan-Bissaka is high on Solskjaer’s shortlist, while Issa Diop, Harry Maguire and Kalidou Koulibaly are centre-half targets.
Daniel James so far remains United’s only signing this summer but reinforcements could be needed in attack, with Romelu Lukaku likely to depart.