Striker Carroll spent a year on loan at Upton Park in the 2012-13 season, before switching permanently from Anfield to London for an estimated £15 million transfer fee.
The 25-year-old has endured several injury spells since, with heel and ankle problems restricting him to just 12 Premier League starts last term - and ruling him out until November this term.
But Allardyce insists every key figure at the club wanted to sign Carroll, and has backed him to hit top form when he returns from his latest lay-off.
"Everyone wanted to sign Andy Carroll when he finished the season, everybody," Allardyce told reporters.
"There wasn't one person from the owners to any members of staff who didn't want to sign him.
"What we have to hope is eventually we will get him fit and he will produce a full season for us. We know what a big player he is for us. We know what effect he has on our team but what effect he has against the opposition.
"When they see him on the team sheet he strikes fear into every defence in the Premier League, no matter how good they are."
Allardyce also believes Carroll has the perfect asset for his rehabilitation - with girlfriend and former reality TV star Billi Mucklow a yoga instructor.
"I think Andy with his girlfriend, as far as I can see, is doing lots of good things," he said.
"I've seen pictures of Andy in the gym she has, which is obviously an exceptionally nice place to be.
"If he's doing some workouts with his girlfriend as well as his rehab with us, it can only benefit and it passes the time. I think for a player the big problem is sitting with a boot on and getting past the time and the boredom. There has to be some release from that, but make sure it's the right release, not the wrong one."
West Ham's Premier League season kicks off on Saturday when they host London rivals Tottenham.