The towering England man has proved a handful for Premier League defences this term having signed a season-long deal at Upton Park - but is scheduled to return to parent club Liverpool this summer.
And Hammers team-mate Cole said: "West Ham definitely want to keep Andy Carroll.
"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 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."
West Ham assistant manager Neil McDonald said on the club's official website: "Andy Carroll is enjoying his time in London, is enjoying playing for West Ham and is enjoying being on the training field and trying to contribute for the team. He's scoring goals and making goals for other people."