Green, 31, will be out of action for around four weeks with a knee cartilage injury.
His unavailability leaves the squad with Peter Kurucz, Marek Stech and Ruud Boffin as options between the posts, although they lack sufficient first-team experience.
"We've been hot on the phone and secured an excellent signing in the short-term in Manuel Almunia from Arsenal," manager Sam Allardyce told West Ham TV.
"It's a big 'thank you' again to Arsene Wenger and Arsenal for allowing us to get such a high-class and experienced goalkeeper to come in and fill in for Robert for the time being."
West Ham return to Championship action tomorrow (Saturday) away to Crystal Palace.