The Liverpool striker is on a season-long loan at Upton Park but former Reds man Cole would love it he could somehow remain in East London.
"West Ham definitely want to keep Andy Carroll," Cole told talkSPORT.
"I don't know the ins and outs of the deal but the way he's been playing in the last couple of weeks you can understand why Liverpool paid £35m for him a couple of years ago.
"I think all the boys want him to stay at West Ham. He's (his own) man, though, and he has his own agenda.
"I don't know what Liverpool are thinking and I'm sure there will be other clubs in for him. West Ham would like to keep him, he's our top scorer. He's got to do what's right for him, though."