The former England midfielder admitted he was frustrated to be hauled off before half-time in West Ham's comprehensive 3-0 defeat to Chelsea on Saturday.
But the 32-year-old rubbished suggestions he is unsettled at the Hammers, and under boss Sam Allardyce.
"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," he told West Ham's official club website.
"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 said West Ham are desperate to shrug off that Chelsea malaise, with a victory when they clash with Fulham in Barclays Premier League action at Upton Park on Saturday.
Cole said: "The manager has been great to me since I came back and supported me while I was out with a couple of frustrating injuries. 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."