The striker has served his three-game suspension and is available again for the Hammers, who have recorded four wins in a row, while Roberto Martinez will be without Phil Jagielka
The 25-year-old has served his three-match ban after receiving a straight red card against Swansea City on February 1.
Carroll was dismissed by referee Howard Webb when he appeared to catch Chico Flores in the face with his arm, and a Football Association tribunal upheld the suspension after West Ham's initial appeal had failed.
Speaking ahead of Saturday's trip to Merseyside, Allardyce put forward Carroll's case for inclusion in England's World Cup squad, even though the 59-year-old was unable to guarantee him a place in his own starting side.
Allardyce's dilemma centres on whether to recall Carroll or to retain faith in Carlton Cole, who has led the line in the ex-Liverpool man's absence and scored in last weekend's 3-1 victory over Southampton.
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The Londoners have scored nine times in that run while conceding just once, and have kept the most clean sheets in the division this season with 13 in 27 outings.
By contrast, Everton have lost three of their last four Premier League matches, albeit with those defeats all coming away and against teams in the top five in the form of Liverpool, Tottenham and Chelsea.
If recent history is anything to go by, however, then the seventh-placed Merseysiders can head into this fixture full of optimism.
Everton have gone 11 top-flight games without defeat against West Ham, and won the reverse fixture 3-2 back in September.
They will be without captain Phil Jagielka (hamstring), while striker Lacina Traore is also absent with a similar issue.
However, Martinez can include striker Romelu Lukaku, who returns from a three-match lay-off with an ankle injury, while Antolin Alcaraz is also back in contention following a muscle problem.
Allardyce, meanwhile, is without on-loan Roma striker Marco Borriello (calf).