Poulter and Casey were leading contenders despite missing the 36-hole cut at this week's PGA Tour's Deutsche Bank Championship in Norton, Mass.
Montgomerie, an eight-time Order of Merit champion on the European Tour, has a 20-9-7 record after playing in every Ryder Cup since 1991. Clarke had made every team since 1997.
"I have to pick two picks," Faldo told Montgomerie in a voice-mail message.
He said he spoke to Clarke by telephone.
The team also includes Padraig Harrington, Sergio Garcia, Lee Westwood, Robert Karlsson, Henrik Stenson, Miguel Angel Jimenez, Graeme McDowell, Justin Rose, Soren Hansen and Oliver Wilson.
The final four members of the United States team will be named Tuesday by Paul Azinger, the captain. Already on the team are Phil Mickelson, Kenny Perry, Jim Furyk, Anthony Kim, Justin Leonard, Stewart Cink, Boo Weekley and Ben Curtis.