The former England midfielder has been linked with a move away from Upton Park after he was substituted before half-time during the Hammers' 3-0 loss to Chelsea last Saturday.
However, the 32-year-old rubbished claims that he is unhappy at West Ham, and under boss Sam Allardyce.
Cole said: "I was as frustrated as anyone after Saturday's game with Chelsea but my only focus at the moment is on getting my head down, working hard and helping the team out of the difficult run we're going through at the moment.
"I'm really enjoying being back at West Ham as it's a club that means so much to me.
"As a squad we've still got a lot to play for this season and I'm determined to play my part in making sure we achieve the goals we set ourselves.
"I'm happy at the club and the only way I'll be leaving is if they tell me they don't want me anymore."
Cole says West Ham are desperate to atone for the home defeat to Chelsea and plan on returning to winning ways against Fulham on Saturday.
"The manager has been great to me since I came back and supported me while I was out with a couple of frustrating injuries," he said.
"I now feel fully fit and I'm ready to help him and the squad whenever I'm needed during a busy schedule in the coming weeks.
"We face a really important game against Fulham on Saturday and I have seen from training this week how much the lads want to win it. If we can get the three points that gives us the perfect platform to go into the away game at Crystal Palace three days later and get a little run going.
"I know the fans will get behind us on Saturday and hopefully we can give them what they all want to see. They're up there with the best in the land when they're on song, so it's up to us to give them something cheer about."