The 24-year-old England international, who spent last season on loan at Upton Park, recently signed a permanent deal with the Hammers for around £15m.
"He's only 24 years old, so he's still got a long way to go to reach his best and hopefully we'll be the club to deliver a better player and a better Andy Carroll in the next few years," Allardyce told talkSPORT.
"The fans love Andy, they love to see players committed to the shirt, they want to see passionate players. It'll be great for him, for England and certainly for West Ham."
Carroll has penned a six-year contract with the club.