The Hammers returned to England's top flight after securing promotion from the Championship last season via a play-off victory against Ian Holloway's Blackpool.
And they hoped to bring in the 23-year-old, who has seemingly been deemed surplus to requirements by Reds boss Brendan Rodgers, on a loan worth £2 million before paying £16m to make the deal permanent.
However, Carroll has rejected the chance to move to east London citing his preference for a transfer to a club competing in Europe, a move which has disappointed Allardyce.
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"We wouldn't have thought we would have had any chance of signing a player of Andy Carroll's capabilities but it became clear a couple of weeks ago there was a possibility of a loan.
"Unfortunately, at this moment, it's not the case."
Hammers midfielder Kevin Nolan has also weighed in on the discussion, telling the Daily Star that he would love to once again line-up alongside his former Newcastle team-mate.
"I would be delighted if Andy would come and join us here," Nolan said.
"I am not too sure whether Brendan Rodgers will let him leave but there is lots of speculation - there always is surrounding Andy.
"If there is a chance, we would be in the reckoning for him because he is such a quality player."