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Rice, who had previously played every minute in the Premier League this season, was absent for the Hammers trip to Wolves on Monday as David Moyes's men look to move fourth in the table.
"It's disappointing. We knew after the England game he wasn't going to be available," said Moyes.
"We've been over the disappointment of that feeling and it gives other people opportunities and that's what squads are for and, hopefully, we've got the squad to deal with it.
"We know we're missing a good player but the team's been playing really well."
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Rice wrote on Twitter: "Rehab has started. Gutted to be missing the game tonight.
"Overwhelmed by all the support but I'll be back in no time."
Sky Sports and the BBC reported Rice will be sidelined for at least four weeks.
The 22-year-old has established himself as a regular starter for England under Gareth Southgate.
Rice has over two months to prove his fitness for Euro 2020 with England's first match against Croatia on June 13.
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