The 23-year-old was taken to hospital yesterday after complaining of stomach pains at the Hammers' Chadwell Heath training ground yesterday.
He underwent an hour-long operation at an Essex hospital and is expected to be back in around a month's time.
Noble has performed admirably in the centre of the park with Scott Parker this term in what has been a tough campaign for the east London outfit.
Avram Grant's team sit three points from safety after their latest defeat at the hands of Arsenal on Saturday.
The Israeli, who replaced Gianfranco Zola at the helm this summer, will be under even more pressure should his side not come away from Saturday's match at Birmingham with at least a point.
Grant is short of options in central midfield having seen Thomas Hitzlsperger ruled out until the new year while Kieron Dyer is struggling to regain full fitness following his previous injury problems.
Radoslav Kovac, who has started only one league match so far this year, is the most likely replacement for Noble, who along with Parker and Robert Green has played every match for the Hammers this year.
Former Chelsea boss Grant will be hoping that defensive duo Matthew Upson and Manuel Da Costa will be available for the trip to St Andrew's.
England centre-back Upson missed Saturday's match with a neck injury while Da Costa picked up a knock to his ankle at the Emirates Stadium but it is thought that both should be fit for the weekend.