The Hammers were keen on the wantaway England international and wanted to recruit him on a one-year loan deal.
However, Liverpool boss Brendan Rodgers has continued to insist that they will entertain no loan offer for Carroll and will listen to only permanent offers.
"I would have probably stuck my neck out and said we would almost certainly stay up with him on board," Allardyce was quoted as saying in The People.
"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."
Carroll has been left of Liverpool's Europa League squad, sparking suggestions that he could still be let go before the summer transfer window shuts down.