The former Liverpool midfielder signed an 18-month deal with the Hammers in an effort to kick-start a stalling career at the club where he first burst on to the scene as a teenager.
Cole, 31, said: "I've had a great career but by no means am I finished.
"There will always be the England thing in the back of my mind because I played 56 times for my country. I haven't played since the World Cup in 2010 so something is missing there.
"I still want to be part of that but you can't talk yourself into England squads, you've got to perform. I want to win things at West Ham and play for England again."