Speaking after the match Moyes told reporters: "Rooney has got a groin strain" and asked if he would be able to face the Hammers he added: "He's got a chance".
The news is a blow to the reigning champions, particularly with fellow striker Robin van Persie set for a month on the sidelines with a thigh strain that rules him out of the traditional busy run of December games.
The Premier League champions face West Ham, Hull City, Norwich City and Tottenham in the space of a 12-day period that starts with Saturday's visit of Sam Allardyce's men to Old Trafford.
Rooney is the side's top scorer in the league this season with eight goals in 14 appearances and has earned regular praise from Moyes for his fine form.
United are eighth in the Premier League, some 10 points adrift of leaders Arsenal.