Rice picked up the injury while on international duty with the England national team against Poland in the FIFA World Cup qualifiers last week.
The 24-year-old, who had been an ever-present for David Moyes’ men in the Premier League this season, missed Monday’s 3-2 win at Wolverhampton Wanders which propelled West Ham into fourth position.
Before the game at Molineux, Rice tweeted: "Rehab has started. Gutted to be missing the game tonight. Overwhelmed by all the support but I'll be back in no time. Come on the boys!"
"It is disappointing," said West Ham manager Moyes ahead of their triumph over Wolves.
"We knew after the England game that he wasn't going to be available. But we move on, we've [got] over the disappointment and that feeling.
"It gives other people opportunities. That's what squads are for and hopefully we've got the squad to deal with it.
"We've moved on from it now. We know we are missing a good player but the team has been playing really well this year."
Rice has made 29 appearances in the top flight for West Ham this season, scoring one goal.
England boss Gareth Southgate will be keeping a close eye on the 22-year-old’s situation, with Rice an integral part of his plans for EURO 2020.
