Allardyce is set to hold talks over his future with the club's hierarchy at the end of the current Premier League campaign, which sees West Ham sitting 10th.
"Everyone loves him, everyone loves the training and the routine we are in," Carroll told the London Evening Standard. "I am sure all the lads want him to stay next season."
West Ham's season started promisingly, but hopes of European qualification have slowly faded away courtesy of a run of two wins in 16 matches.
"The manager is strong enough to deal with that and I think so are all the players," he added. "It is just unlucky the way the games have been going.
"In the last minute of the game the other day [against Stoke], they equalised, so we've just been very unlucky.
"We need to start pushing for Europe next season. We started off well this season and need to maintain it all the way through."
A knee injury has kept Carroll sidelined since February, but the former Newcastle United and Liverpool man believes he has an explanation for his persistent fitness issues.
"It is just the way I play - aggressive play," he added. "You see the way I did my knee. I went up for a header, went to run in where one of the defenders was standing and I've clattered him in the back and I've ended up doing my own injury.
"That's just me. Most of the injuries - I've hyper-extended when running for a tackle - it is just because I am trying to give 100 per cent every time and throw myself into everything. I think that is what happens."
