Carroll, whose season-long loan from Liverpool is due to expire shortly, has been in good form recently and is a key part of Sam Allardyce's side.
Cole, who rejoined West Ham from Liverpool in January, is working his way back to fitness and wants England forward Carroll to sign up in the summer.
"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 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."
It remains to be seen if a permanent deal can be thrashed out between the two clubs and if Carroll would be interested in a full-time switch to east London.
Allardyce has been non-committal about securing Carroll's services beyond the end of the current campaign but Hammers' assistant manager Neil McDonald revealed the 24-year-old is happy at the club.
"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 told the club's official website.
"He's scoring goals and making goals for other people."
Carroll currently sits top of the West Ham scoring charts, level on seven goals with captain Kevin Nolan, despite missing parts of the season with hamstring and knee injuries.