The former Chelsea striker's future has been questioned following the signing of the Liverpool man, but Cole took to Twitter to reveal that he has no intention of moving on from the Hammers.
"Apparently I'm leaving, ur avvin a laff aint ya!" Tweeted Cole.
"Cos a striker comes in doesn't mean I have to leave."
The 28-year-old believes that Carroll's presence in the squad will only help improve the side.
"Andy Carroll is a good addition to the squad and hopefully he makes a positive impact to our side. Onwards and upwards," he added.
"Competition for places is good for players and the team, you need it in every position. I feel it's gonna be a good season for the Hammers."
West Ham host Fulham in the Premier League tomorrow.