Cole has yet to score this season and has fallen down the pecking order at West Ham following the arrival of Andy Carroll during the summer.
The 28-year-old is out of contract at the end of the season but is adamant that he sees his long-term future with the Hammers.
He told The Sun: "Because Andy Carroll is here, it doesn't mean I am going to go.
"I don't want to leave. I've told the board that I want to stay for the rest of my career. I want to stay and help out the club anyway I can.
"I've never been one to lie down. Andy's a great player, I see it as good competition having him here and it will be good for the club."